Introduction
For a connoisseur of positivity, the Riemann Ξ \Xi Ξ
We use the usual normalization (Titchmarsh, [32 ] , Section 2.1) for Ξ \Xi Ξ
ξ ( s ) = 1 2 s ( s − 1 ) π − s / 2 Γ ( s / 2 ) ζ ( s ) , c 0 = ξ ( 1 2 ) > 0 , Ξ ( z ) = ξ ( 1 2 + i z ) , (1) \xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad c_0=\xi\left(\frac{1}{2}\right)>0,\qquad\Xi(z)=\xi\left(\frac{1}{2}+iz\right), \tag*{(1)} ξ ( s ) = 2 1 s ( s − 1 ) π − s /2 Γ ( s /2 ) ζ ( s ) , c 0 = ξ ( 2 1 ) > 0 , Ξ ( z ) = ξ ( 2 1 + i z ) , (1) and define the following functions
F ( z ) : = Ξ ( z ) / c 0 , h ( z ) : = ξ ( 1 2 + z ) / c 0 , B ( s ) : = log h ( s ) ( s ≥ 0 ) . (2) F(z):=\Xi(z)/c_0,\qquad h(z):=\xi\left(\frac{1}{2}+\sqrt{z}\right)/c_0,\qquad B(s):=\log h(s)\quad(s\geq0). \tag*{(2)} F ( z ) := Ξ ( z ) / c 0 , h ( z ) := ξ ( 2 1 + z ) / c 0 , B ( s ) := log h ( s ) ( s ≥ 0 ) . (2) Let Z + \mathcal{Z}_+ Z + ρ = β + i γ \rho=\beta+i\gamma ρ = β + iγ γ > 0 \gamma>0 γ > 0
H ( t ) : = ∑ ρ ∈ Z + e − t ( γ − i ( β − 1 2 ) ) 2 ( t > 0 ) . H(t):=\sum_{\rho\in\mathcal{Z}_+}e^{-t(\gamma-i(\beta-\frac{1}{2}))^2}\qquad(t>0). H ( t ) := ρ ∈ Z + ∑ e − t ( γ − i ( β − 2 1 ) ) 2 ( t > 0 ) . Under RH three positivity properties hold: (a) H H H completely monotone (i.e., ( − 1 ) n H ( n ) ≥ 0 (-1)^nH^{(n)}\geq0 ( − 1 ) n H ( n ) ≥ 0 [30 ] ; Gröchenig, [11 ] ), $1/(c_0h) and 1 / Ξ 1/\Xi 1/Ξ Ξ \Xi Ξ R \mathbb{R} R [7 ] , Theorem 2.9). Without RH, an off-line zero ρ \rho ρ 1 − ρ ˉ 1-\bar{\rho} 1 − ρ ˉ Z + \mathcal{Z}_+ Z + H H H H H H t t t
Our main results are the following. Items marked * overlap in part with earlier or parallel work; the precise relation is described in Section 5 and in the discussion of the corresponding result.
(1)* Let Θ \Theta Θ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Λ \Lambda Λ R \mathbb{R} R 1 / ( c 0 h ) 1/(c_0h) 1/ ( c 0 h ) 1 / Ξ 1/\Xi 1/Ξ α > 0 \alpha>0 α > 0 h − α h^{-\alpha} h − α [ c 0 / Ξ ( 1 2 + ∥ u ∥ ) ] α \left[c_0/\Xi\left(\frac{1}{2}+\lVert u\rVert\right)\right]^\alpha [ c 0 /Ξ ( 2 1 + ∥ u ∥ ) ] α R d \mathbb{R}^d R d d d d [26 ] was answered by Polson [24 ] , v24, Theorem A; the proof here uses zeros only through height 100 and adds the regularity of the densities. This result answers a positive-definiteness question raised by Gröchenig [11 ] .
(2) The densities Θ \Theta Θ Λ \Lambda Λ Θ , Λ ∈ P F 2 \Theta,\Lambda\in\mathrm{PF}_2 Θ , Λ ∈ PF 2 3 ⋅ 10 12 3\cdot10^{12} 3 ⋅ 1 0 12 Λ ∈ P F 10 \Lambda\in\mathrm{PF}_{10} Λ ∈ PF 10 P F k \mathrm{PF}_k PF k k k k
(3)* If RH holds up to height T > 1 2 T>\frac{1}{2} T > 2 1 Ξ \Xi Ξ L n [ Ξ ] ( x ) ≥ 0 L_n[\Xi](x)\geq0 L n [ Ξ ] ( x ) ≥ 0 n n n ∣ x ∣ ≤ T − 1 2 \lvert x\rvert\leq T-\frac{1}{2} ∣ x ∣ ≤ T − 2 1 L n [ Ξ ] ( x ) L_n[\Xi](x) L n [ Ξ ] ( x ) y 2 n y^{2n} y 2 n ∣ Ξ ( x + i y ) ∣ 2 \lvert\Xi(x+iy)\rvert^2 ∣ Ξ ( x + i y ) ∣ 2 n = 1 n=1 n = 1 [8 ] .
The inputs to our results are the critical strip, explicit zero counting, and published zero verification. Our conclusions do not extend the verified height, and they are merely weak necessary conditions. The global Laguerre inequality L 1 [ Ξ ] ≥ 0 L_1[\Xi]\geq0 L 1 [ Ξ ] ≥ 0
Construction of the reciprocal laws and
Assuming RH, the densities Θ \Theta Θ Λ \Lambda Λ P F ∞ \mathrm{PF}_\infty PF ∞ [30 ] , Theorems 1 and 2), specialized to Ξ \Xi Ξ [11 ] , Theorems 3–5. But without RH, at least their existence follows easily. Using the notation from (1.1) and (1.2), from Hadamard’s product for h h h B ′ ( s ) = ∫ 0 ∞ e − s t H ( t ) d t B'(s)=\int_0^\infty e^{-st}H(t)\,\mathrm{d}t B ′ ( s ) = ∫ 0 ∞ e − s t H ( t ) d t H > 0 H>0 H > 0 B B B h − α = e − α B h^{-\alpha}=e^{-\alpha B} h − α = e − α B
Theorem 1.1 (Reciprocal xi and probability laws). For t > 0 t>0 t > 0 H ( t ) > 0 H(t)>0 H ( t ) > 0 H ′ ( t ) < 0 H'(t)<0 H ′ ( t ) < 0 B B B
B ( s ) = ∫ 0 ∞ ( 1 − e − s t ) H ( t ) t d t . (3) B(s)=\int_0^\infty(1-e^{-st})\frac{H(t)}{t}\,\mathrm{d}t. \tag*{(3)} B ( s ) = ∫ 0 ∞ ( 1 − e − s t ) t H ( t ) d t . (3) Moreover, for every α > 0 \alpha>0 α > 0 S α S_\alpha S α E e − s S α = h ( s ) − α \mathbb{E}e^{-sS_\alpha}=h(s)^{-\alpha} E e − s S α = h ( s ) − α C ∞ C^\infty C ∞ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) d ≥ 1 d\geq1 d ≥ 1
h ( ∥ u ∥ 2 ) − α = c 0 α Ξ ( 1 2 + ∥ u ∥ ) − α (4) h(\lVert u\rVert^2)^{-\alpha}=c_0^\alpha\Xi\left(\frac{1}{2}+\lVert u\rVert\right)^{-\alpha} \tag*{(4)} h (∥ u ∥ 2 ) − α = c 0 α Ξ ( 2 1 + ∥ u ∥ ) − α (4) is strictly positive definite on R d \mathbb{R}^d R d R d \mathbb{R}^d R d C d \mathbb{C}^d C d
Log-concavity and finite PF orders
As noted, RH implies Θ \Theta Θ Λ \Lambda Λ P F ∞ \mathrm{PF}_\infty PF ∞ P F m \mathrm{PF}_m PF m det [ f ( x i − y j ) ] i , j = 1 m ≥ 0 \det[f(x_i-y_j)]_{i,j=1}^m\geq0 det [ f ( x i − y j ) ] i , j = 1 m ≥ 0 x 1 < ⋯ < x m x_1<\cdots<x_m x 1 < ⋯ < x m y 1 < ⋯ < y m y_1<\cdots<y_m y 1 < ⋯ < y m P F 2 \mathrm{PF}_2 PF 2
Theorem 1.2 (Log-concavity and PFk _k k The densities Θ \Theta Θ Λ \Lambda Λ P F 2 \mathrm{PF}_2 PF 2 ( log Θ ) ′ ′ ( x ) < 0 (\log\Theta)''(x)<0 ( log Θ ) ′′ ( x ) < 0 x > 0 x>0 x > 0 ( log Λ ) ′ ′ ( x ) < 0 (\log\Lambda)''(x)<0 ( log Λ ) ′′ ( x ) < 0 x ∈ R x\in\mathbb{R} x ∈ R Λ ∈ P F 10 \Lambda\in\mathrm{PF}_{10} Λ ∈ PF 10
Generalized Laguerre inequalities for
Ξ \Xi Ξ Ξ \Xi Ξ R \mathbb{R} R f f f
L n [ f ] ( x ) : = 1 ( 2 n ) ! ∑ j = 0 2 n ( − 1 ) j + n ( 2 n j ) f ( j ) ( x ) f ( 2 n − j ) ( x ) , n ∈ N 0 . (5) L_n[f](x):=\frac{1}{(2n)!}\sum_{j=0}^{2n}(-1)^{j+n}\binom{2n}{j}f^{(j)}(x)f^{(2n-j)}(x),\qquad n\in\mathbb{N}_0. \tag*{(5)} L n [ f ] ( x ) := ( 2 n )! 1 j = 0 ∑ 2 n ( − 1 ) j + n ( j 2 n ) f ( j ) ( x ) f ( 2 n − j ) ( x ) , n ∈ N 0 . (5) These are the coefficients of y 2 n y^{2n} y 2 n ∣ f ( x + i y ) ∣ 2 |f(x+iy)|^2 ∣ f ( x + i y ) ∣ 2 L 0 [ f ] = f 2 L_0[f]=f^2 L 0 [ f ] = f 2 L 1 [ f ] = ( f ′ ) 2 − f f ′ ′ L_1[f]=(f')^2-ff'' L 1 [ f ] = ( f ′ ) 2 − f f ′′ L n [ f ] ≥ 0 L_n[f]\ge0 L n [ f ] ≥ 0 n ≥ 0 n\ge0 n ≥ 0 f f f L n [ Ξ ] ≥ 0 L_n[\Xi]\ge0 L n [ Ξ ] ≥ 0 [8 ] proved L 1 [ Ξ ] ≥ 0 L_1[\Xi]\ge0 L 1 [ Ξ ] ≥ 0 [6 ] ); we extend that result to show that L n [ Ξ ] ≥ 0 L_n[\Xi]\ge0 L n [ Ξ ] ≥ 0
Theorem 1.3 (Generalized Laguerre on an interval). Let T > 1 / 2 T>1/2 T > 1/2 β + i γ \beta+i\gamma β + iγ 0 < γ ≤ T 0<\gamma\le T 0 < γ ≤ T β = 1 / 2 \beta=1/2 β = 1/2
L n [ Ξ ] ( x ) ≥ 0 ( n = 0 , 1 , 2 , … , ∣ x ∣ ≤ T − 1 2 ) . L_n[\Xi](x)\ge0 \qquad(n=0,1,2,\ldots,\quad|x|\le T-\frac{1}{2}). L n [ Ξ ] ( x ) ≥ 0 ( n = 0 , 1 , 2 , … , ∣ x ∣ ≤ T − 2 1 ) . Moreover, L n [ Ξ ] ( x ) > 0 L_n[\Xi](x)>0 L n [ Ξ ] ( x ) > 0 x x x Ξ \Xi Ξ n n n
By Platt and Trudgian [20 ] the critical line hypothesis holds for T = T ∗ : = 3 ⋅ 10 12 T=T_*:=3\cdot10^{12} T = T ∗ := 3 ⋅ 1 0 12 L n [ Ξ ] ≥ 0 L_n[\Xi]\ge0 L n [ Ξ ] ≥ 0 ∣ x ∣ ≤ T ∗ − 1 2 |x|\le T_*-\frac{1}{2} ∣ x ∣ ≤ T ∗ − 2 1 n = 1 n=1 n = 1 [8 ] , who assume that the zeros are simple; see also Csordas et al. [9 ] , Csordas (1998, remark after Problem 2) and Csordas [6 ] .
Proof of Theorem 1.1: the reciprocal laws
This section proves Theorem 1.1 . The construction needs two analytic facts: the canonical product must yield a convergent Laplace integral, and H H H § 2.3 and 2.4 then construct the laws and the densities Θ \Theta Θ Λ \Lambda Λ
Analytic and numerical inputs
We use the classical analytic continuation, functional equation, zero symmetries, and critical strip of ζ \zeta ζ [32 ] ). In particular δ ρ : = β − 1 2 \delta_\rho:=\beta-\frac{1}{2} δ ρ := β − 2 1 ∣ δ ρ ∣ < 1 / 2 |\delta_\rho|<1/2 ∣ δ ρ ∣ < 1/2 ξ \xi ξ c 0 > 0 c_0>0 c 0 > 0
Let N ( Y ) N(Y) N ( Y ) Y Y Y
M ( Y ) = Y 2 π ( log Y 2 π − 1 ) , M ( 0 ) = 0. M(Y)=\frac{Y}{2\pi}\left(\log\frac{Y}{2\pi}-1\right),\qquad M(0)=0. M ( Y ) = 2 π Y ( log 2 π Y − 1 ) , M ( 0 ) = 0. We use the following consequence of Bellotti and Wong [2 ] :
∣ N ( Y ) − M ( Y ) ∣ ≤ 1 8 log ( Y + 2 ) + 4 ( Y ≥ 0 ) . (6) |N(Y)-M(Y)|\le\frac{1}{8}\log(Y+2)+4 \qquad(Y\ge0). \tag*{(6)} ∣ N ( Y ) − M ( Y ) ∣ ≤ 8 1 log ( Y + 2 ) + 4 ( Y ≥ 0 ) . (6) For all U ≥ 100 U\ge100 U ≥ 100 (6) implies
N ( Y ) ≤ Y log Y ( Y ≥ U ) , ∑ γ > U γ − 2 ≤ 2 ( log U + 1 ) U < 1. (7) N(Y)\le Y\log Y \qquad(Y\ge U),\qquad\sum_{\gamma>U}\gamma^{-2}\le\frac{2(\log U+1)}{U}<1. \tag*{(7)} N ( Y ) ≤ Y log Y ( Y ≥ U ) , γ > U ∑ γ − 2 ≤ U 2 ( log U + 1 ) < 1. (7) The sum estimate follows by Stieltjes integration by parts, dropping the nonpositive boundary term. The proved finite zero inputs are
γ ≤ T ∗ ⟹ δ ρ = 0 , 14 < γ 1 < 15 , (8) \gamma\le T_* \Longrightarrow\delta_\rho=0,\qquad14<\gamma_1<15, \tag*{(8)} γ ≤ T ∗ ⟹ δ ρ = 0 , 14 < γ 1 < 15 , (8) where γ 1 \gamma_1 γ 1 [20 ] , which verifies height 3 000 175 332 800 3\,000\,175\,332\,800 3 000 175 332 800 Theorem 1.1 and the log-concavity in Theorem 1.2 . Table 1 lists the numerical inputs of each result; the certificates are in Appendix § 10 .
For ρ ∈ Z + \rho\in\mathbb{Z}_+ ρ ∈ Z + λ ρ : = ( γ − i δ ρ ) 2 \lambda_\rho:=(\gamma-i\delta_\rho)^2 λ ρ := ( γ − i δ ρ ) 2 H ( t ) = ∑ ρ ∈ Z + e − t λ ρ H(t)=\sum_{\rho\in\mathbb{Z}_+}e^{-t\lambda_\rho} H ( t ) = ∑ ρ ∈ Z + e − t λ ρ
Lemma 2.1 (Product and convergence). The function h h h 1 / 2 1/2 1/2
h ( z ) = ∏ ρ ∈ Z + ( 1 + z λ ρ ) . (9) h(z)=\prod_{\rho\in\mathbb{Z}_{+}}\left(1+\frac{z}{\lambda_\rho}\right). \tag*{(9)} h ( z ) = ρ ∈ Z + ∏ ( 1 + λ ρ z ) . (9) The product converges locally uniformly. If a ρ = Re λ ρ a_\rho=\operatorname{Re}\lambda_\rho a ρ = Re λ ρ a ρ > 0 a_\rho>0 a ρ > 0 ∑ ρ a ρ − 1 < ∞ \sum_\rho a_\rho^{-1}<\infty ∑ ρ a ρ − 1 < ∞ h ( s ) > 0 h(s)>0 h ( s ) > 0 s ≥ 0 s\geq0 s ≥ 0 H H H Re t > 0 \operatorname{Re}t>0 Re t > 0
Proof. Evenness turns the order-one Xi function into an order-1 / 2 1/2 1/2 ρ , 1 − ρ \rho,1-\rho ρ , 1 − ρ Z + \mathbb{Z}_{+} Z + (9) , with constant one. For large ordinates a ρ = γ 2 − δ ρ 2 ≥ γ 2 − 1 / 4 a_\rho=\gamma^2-\delta_\rho^2\geq\gamma^2-1/4 a ρ = γ 2 − δ ρ 2 ≥ γ 2 − 1/4 (8) makes a ρ = γ 2 > 0 a_\rho=\gamma^2>0 a ρ = γ 2 > 0 (9) combine into positive squared moduli on the positive axis; real factors are positive. On a compact subset of Re t > 0 \operatorname{Re}t>0 Re t > 0 e − c γ 2 + C γ e^{-c\gamma^2+C\gamma} e − c γ 2 + C γ γ \gamma γ
Positivity of H H H − H ′ -H' − H ′
A high zero can oscillate only after its exponential weight has become small. The estimate below makes this delay quantitative for both H H H − H ′ -H' − H ′ [24 ] and Polson [23 ] ; the estimate below makes it valid for every t > 0 t>0 t > 0
Put q = γ 1 2 ∈ ( 196 , 225 ) q=\gamma_1^2\in(196,225) q = γ 1 2 ∈ ( 196 , 225 )
a ρ = γ 2 − δ ρ 2 , b ρ = 2 γ δ ρ , λ ρ = a ρ − i b ρ . a_\rho=\gamma^2-\delta_\rho^2,\qquad b_\rho=2\gamma\delta_\rho,\qquad\lambda_\rho=a_\rho-i b_\rho. a ρ = γ 2 − δ ρ 2 , b ρ = 2 γ δ ρ , λ ρ = a ρ − i b ρ . Then ∣ b ρ ∣ ≤ γ |b_\rho|\leq\gamma ∣ b ρ ∣ ≤ γ ∣ λ ρ ∣ = γ 2 + δ ρ 2 |\lambda_\rho|=\gamma^2+\delta_\rho^2 ∣ λ ρ ∣ = γ 2 + δ ρ 2
Lemma 2.2 (One retained real zero). Suppose 100 ≤ U ≤ T ∗ 100\leq U\leq T_* 100 ≤ U ≤ T ∗
H > U ( t ) = ∑ γ > U e − t λ ρ , K U ( t ) = e − q t + H > U ( t ) , ε U = U 4 e − U . H_{>U}(t)=\sum_{\gamma>U}e^{-t\lambda_\rho},\qquad K_U(t)=e^{-qt}+H_{>U}(t),\qquad\varepsilon_U=U^4e^{-U}. H > U ( t ) = γ > U ∑ e − t λ ρ , K U ( t ) = e − q t + H > U ( t ) , ε U = U 4 e − U . For every t > 0 t>0 t > 0
K U ( t ) ≥ ( 1 − ε U ) e − q t > 0 , − K U ′ ( t ) ≥ ( 1 − ε U ) q e − q t > 0. K_U(t)\geq(1-\varepsilon_U)e^{-qt}>0,\qquad-K'_U(t)\geq(1-\varepsilon_U)qe^{-qt}>0. K U ( t ) ≥ ( 1 − ε U ) e − q t > 0 , − K U ′ ( t ) ≥ ( 1 − ε U ) q e − q t > 0. In particular H > 0 H>0 H > 0 − H ′ > 0 -H'>0 − H ′ > 0
Proof. The real summands of H H H − H ′ -H' − H ′
e − a ρ t cos ( b ρ t ) , e − a ρ t [ a ρ cos ( b ρ t ) + b ρ sin ( b ρ t ) ] . e^{-a_\rho t}\cos(b_\rho t),\qquad e^{-a_\rho t}\left[a_\rho\cos(b_\rho t)+b_\rho\sin(b_\rho t)\right]. e − a ρ t cos ( b ρ t ) , e − a ρ t [ a ρ cos ( b ρ t ) + b ρ sin ( b ρ t ) ] . Both are nonnegative when ∣ b ρ ∣ t ≤ π / 2 |b_\rho|t\leq\pi/2 ∣ b ρ ∣ t ≤ π /2 t > π / ( 2 γ ) t>\pi/(2\gamma) t > π / ( 2 γ ) γ ≥ 100 \gamma\geq100 γ ≥ 100 a ρ − q ≥ γ 2 − 1 / 4 − 225 ≥ 3 γ 2 / 4 a_\rho-q\geq\gamma^2-1/4-225\geq3\gamma^2/4 a ρ − q ≥ γ 2 − 1/4 − 225 ≥ 3 γ 2 /4
e − ( a ρ − q ) t ≤ e − 3 π γ / 8 ≤ e − γ . e^{-(a_\rho-q)t}\leq e^{-3\pi\gamma/8}\leq e^{-\gamma}. e − ( a ρ − q ) t ≤ e − 3 π γ /8 ≤ e − γ . Its negative part is consequently at most e − q t e − γ e^{-qt}e^{-\gamma} e − q t e − γ q e − q t γ 2 e − γ qe^{-qt}\gamma^2e^{-\gamma} q e − q t γ 2 e − γ ∣ λ ρ ∣ ≤ q γ 2 |\lambda_\rho|\leq q\gamma^2 ∣ λ ρ ∣ ≤ q γ 2 (7) and monotonicity of y 2 e − y y^2e^{-y} y 2 e − y y 4 e − y y^4e^{-y} y 4 e − y y ≥ 100 y\geq100 y ≥ 100
∑ γ > U e − γ < U 2 e − U , ∑ γ > U γ 2 e − γ < U 4 e − U . \sum_{\gamma>U}e^{-\gamma}<U^2e^{-U},\qquad\sum_{\gamma>U}\gamma^2e^{-\gamma}<U^4e^{-U}. γ > U ∑ e − γ < U 2 e − U , γ > U ∑ γ 2 e − γ < U 4 e − U . Subtract from the retained terms e − q t e^{-qt} e − q t q e − q t qe^{-qt} q e − q t H − K U H-K_U H − K U H H H
The Bernstein representation and the laws
We first derive the Lévy representation of B B B log ( 1 + s / λ ) = ∫ 0 ∞ ( 1 − e − s t ) e − λ t d t / t \log(1+s/\lambda)=\int_{0}^{\infty}(1-e^{-st})e^{-\lambda t}\,\mathrm{d}t/t log ( 1 + s / λ ) = ∫ 0 ∞ ( 1 − e − s t ) e − λ t d t / t λ \lambda λ λ ρ > 0 \lambda_\rho>0 λ ρ > 0 [4 ] , Chapter 3). Without RH the exponents λ ρ \lambda_\rho λ ρ H ( t ) / t H(t)/t H ( t ) / t [24 ] . Lemma 2.4 supplies smooth positive densities and negative moments. Gaussian subordination gives the radial characteristic functions, and the growth of B B B
The integral characterization of Bernstein functions and the complete Bernstein subclass are described in Schilling et al. [28 ] ; here a Poisson random measure constructs the laws directly.
Proposition 2.3 (Bernstein representation). Equation (3) holds, and for s ≥ 0 s\geq0 s ≥ 0 n ≥ 0 n\geq0 n ≥ 0
( − 1 ) n B ( n + 1 ) ( s ) = ∫ 0 ∞ t n e − s t H ( t ) d t > 0. (-1)^n B^{(n+1)}(s)=\int_{0}^{\infty}t^n e^{-st}H(t)\,\mathrm{d}t>0. ( − 1 ) n B ( n + 1 ) ( s ) = ∫ 0 ∞ t n e − s t H ( t ) d t > 0. Moreover ∫ 0 ∞ ∣ H ( t ) ∣ d t < ∞ \int_{0}^{\infty}|H(t)|\,\mathrm{d}t<\infty ∫ 0 ∞ ∣ H ( t ) ∣ d t < ∞
Proof. Logarithmic differentiation of (9) yields B ′ ( s ) = ∑ ρ ( s + λ ρ ) − 1 B'(s)=\sum_\rho(s+\lambda_\rho)^{-1} B ′ ( s ) = ∑ ρ ( s + λ ρ ) − 1
∑ ρ ∫ 0 ∞ ∣ e − ( s + λ ρ ) t ∣ d t = ∑ ρ ( s + a ρ ) − 1 < ∞ . \sum_\rho\int_{0}^{\infty}\left|e^{-(s+\lambda_\rho)t}\right|\,\mathrm{d}t=\sum_\rho(s+a_\rho)^{-1}<\infty. ρ ∑ ∫ 0 ∞ e − ( s + λ ρ ) t d t = ρ ∑ ( s + a ρ ) − 1 < ∞. Thus B ′ ( s ) = ∫ e − s t H ( t ) d t B'(s)=\int e^{-st}H(t)\,\mathrm{d}t B ′ ( s ) = ∫ e − s t H ( t ) d t ∫ ∣ H ∣ ≤ ∑ ρ a ρ − 1 < ∞ \int|H|\leq\sum_\rho a_\rho^{-1}<\infty ∫ ∣ H ∣ ≤ ∑ ρ a ρ − 1 < ∞ s s s B ( 0 ) = 0 B(0)=0 B ( 0 ) = 0 (3) . The positivity follows from Lemma 2.2; differentiation is justified by integrability at zero and exponential decay at infinity. The Lévy condition follows from ∫ ( 1 ∧ t ) H ( t ) d t / t < ∞ \int(1\wedge t)H(t)\,\mathrm{d}t/t<\infty ∫ ( 1 ∧ t ) H ( t ) d t / t < ∞
For α > 0 \alpha>0 α > 0 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) α H ( t ) d t / t \alpha H(t)\,\mathrm{d}t/t α H ( t ) d t / t S α S_\alpha S α α ∫ H ( t ) d t < ∞ \alpha\int H(t)\,\mathrm{d}t<\infty α ∫ H ( t ) d t < ∞
E e − s S α = e − α B ( s ) = h ( s ) − α . (10) \mathbb{E}e^{-sS_\alpha}=e^{-\alpha B(s)}=h(s)^{-\alpha}. \tag*{(10)} E e − s S α = e − α B ( s ) = h ( s ) − α . (10) Independent sums add the parameter α \alpha α 0 < c < 1 0<c<1 0 < c < 1
B ( s ) − B ( c s ) = ∫ 0 ∞ ( 1 − e − s t ) H ( t ) − H ( t / c ) t d t . (11) B(s)-B(cs)=\int_{0}^{\infty}(1-e^{-st})\frac{H(t)-H(t/c)}{t}\,\mathrm{d}t. \tag*{(11)} B ( s ) − B ( cs ) = ∫ 0 ∞ ( 1 − e − s t ) t H ( t ) − H ( t / c ) d t . (11) Since H H H ( H ( t ) − H ( t / c ) ) / t (H(t)-H(t/c))/t ( H ( t ) − H ( t / c )) / t (11) is nonnegative. Equation (11) therefore constructs an independent R c ≥ 0 R_c\geq0 R c ≥ 0 S α = d c S α ′ + R c S_\alpha\mathrel{\overset{d}{=}}cS'_\alpha+R_c S α = d c S α ′ + R c [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) k ( t ) / t k(t)/t k ( t ) / t k k k [27 ] ). For the xi function see also Polson [24 ] .
Lemma 2.4 (Density, support, and moments). For every α > 0 \alpha>0 α > 0 S α S_\alpha S α C ∞ ( R ) C^\infty(\mathbb{R}) C ∞ ( R ) θ α \theta_\alpha θ α ( − ∞ , 0 ] (-\infty,0] ( − ∞ , 0 ] ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Re s > − q \operatorname{Re}s>-q Re s > − q r < q r<q r < q r ≥ q r\geq q r ≥ q
The proof extracts gamma convolution factors of arbitrarily high order from the Lévy measure. This gives smoothness and full positive support; the remaining moment estimates are in Appendix B.§ 4 . The exponential-moment threshold is also in Polson [24 ] and Polson [23 ] .
Let G d G_d G d S α S_\alpha S α X α , d = 2 S α G d X_{\alpha,d} = \sqrt{2}S_\alpha G_d X α , d = 2 S α G d (4) and density
p α , d ( x ) = ∫ 0 ∞ ( 4 π u ) − d / 2 e − ∥ x ∥ 2 / ( 4 u ) θ α ( u ) d u > 0. p_{\alpha,d}(x) = \int_0^\infty(4\pi u)^{-d/2}e^{-\lVert x\rVert^2/(4u)}\theta_\alpha(u)\,\mathrm{d}u > 0. p α , d ( x ) = ∫ 0 ∞ ( 4 π u ) − d /2 e − ∥ x ∥ 2 / ( 4 u ) θ α ( u ) d u > 0. Its infinite divisibility follows by adding the parameter α \alpha α (11) with c 2 c^2 c 2 c c c [29 ] ; see also Roynette and Yor [26 ] . For the xi function with α = d = 1 \alpha=d=1 α = d = 1 [24 ] . The Gaussian Gram matrix on distinct points is positive definite for each u > 0 u>0 u > 0 θ α \theta_\alpha θ α (4) . By (B.5), its characteristic function is integrable against e A ∥ t ∥ e^{A\lVert t\rVert} e A ∥ t ∥ A > 0 A>0 A > 0 p α , d p_{\alpha,d} p α , d C d \mathbb{C}^d C d
Theorem 1.1 also removes the hypothesis from Roynette and Yor [26 ] , which they prove under RH: for every α > 1 2 \alpha>\frac{1}{2} α > 2 1 H ~ α ≥ 0 \widetilde H_\alpha\geq0 H α ≥ 0
E e − λ 2 H ~ α / 2 = ξ ( α ) ξ ( α + λ ) ( λ ≥ 0 ) . (12) \mathbb{E}e^{-\lambda^2\widetilde H_\alpha/2}=\frac{\xi(\alpha)}{\xi(\alpha+\lambda)}
\qquad(\lambda\geq0). \tag*{(12)} E e − λ 2 H α /2 = ξ ( α + λ ) ξ ( α ) ( λ ≥ 0 ) . (12) Indeed, H ~ = 2 S 1 \widetilde H=2S_1 H = 2 S 1 E e − β 2 H ~ / 2 = c 0 / ξ ( 1 2 + β ) \mathbb{E}e^{-\beta^2\widetilde H/2}=c_0/\xi(\frac{1}{2}+\beta) E e − β 2 H /2 = c 0 / ξ ( 2 1 + β ) β ≥ 0 \beta\geq0 β ≥ 0 [26 ] , applied to H ~ \widetilde H H α − 1 2 > 0 \alpha-\frac{1}{2}>0 α − 2 1 > 0 (12) . For α ≥ 1 \alpha\geq1 α ≥ 1 H ~ α / 2 \widetilde H_\alpha/2 H α /2 [23 ] . The question (V.35) of Roynette and Yor, which asks for the additional factor exp ( λ ξ ′ ( α ) / ξ ( α ) ) \exp(\lambda\xi'(\alpha)/\xi(\alpha)) exp ( λ ξ ′ ( α ) / ξ ( α )) (12) , is not settled by this argument; see Polson [23 ] .
The densities Θ \Theta Θ Λ \Lambda Λ
In the notation of Gröchenig [11 ] , Ξ ( z ) = Ξ 1 ( − z 2 ) \Xi(z)=\Xi_1(-z^2) Ξ ( z ) = Ξ 1 ( − z 2 ) Ξ 1 = c 0 h \Xi_1=c_0h Ξ 1 = c 0 h
Θ ( u ) = θ 1 ( u ) c 0 , Λ ( x ) = p 1 , 1 ( x ) c 0 = 1 c 0 ∫ 0 ∞ e − x 2 / ( 4 u ) 4 π u θ 1 ( u ) d u . (13) \Theta(u)=\frac{\theta_1(u)}{c_0},\qquad\Lambda(x)=\frac{p_{1,1}(x)}{c_0}=\frac{1}{c_0}\int_0^\infty\frac{e^{-x^2/(4u)}}{\sqrt{4\pi u}}\theta_1(u)\,\mathrm{d}u. \tag*{(13)} Θ ( u ) = c 0 θ 1 ( u ) , Λ ( x ) = c 0 p 1 , 1 ( x ) = c 0 1 ∫ 0 ∞ 4 π u e − x 2 / ( 4 u ) θ 1 ( u ) d u . (13) Both integrate to 1 / c 0 1/c_0 1/ c 0
Corollary 2.5 (Positive versions of the two Laplace representations). The density Θ \Theta Θ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Λ \Lambda Λ R \mathbb{R} R
1 c 0 h ( s ) = ∫ 0 ∞ e − s u Θ ( u ) d u , Re s > − γ 1 2 , \frac{1}{c_0h(s)}=\int_0^\infty e^{-su}\Theta(u)\,\mathrm{d}u,\qquad\operatorname{Re}s>-\gamma_1^2, c 0 h ( s ) 1 = ∫ 0 ∞ e − s u Θ ( u ) d u , Re s > − γ 1 2 , 1 Ξ ( z ) = ∫ R e − z x Λ ( x ) d x , ∣ Re z ∣ < γ 1 . (14) \frac{1}{\Xi(z)}=\int_{\mathbb{R}}e^{-zx}\Lambda(x)\,\mathrm{d}x,\qquad|\operatorname{Re}z|<\gamma_1. \tag*{(14)} Ξ ( z ) 1 = ∫ R e − z x Λ ( x ) d x , ∣ Re z ∣ < γ 1 . (14) In particular, for arbitrary c j ∈ C c_j\in\mathbb{C} c j ∈ C t j ∈ R t_j\in\mathbb{R} t j ∈ R
∑ j , k c j c k ‾ ξ ( 1 2 + t j − t k ) = ∫ R ∣ ∑ j c j e i t j x ∣ 2 Λ ( x ) d x ≥ 0. (15) \sum_{j,k}\frac{c_j\overline{c_k}}{\xi(\frac{1}{2}+t_j-t_k)}
=\int_{\mathbb{R}}\left|\sum_j c_j e^{it_jx}\right|^2\Lambda(x)\,\mathrm{d}x\geq0. \tag*{(15)} j , k ∑ ξ ( 2 1 + t j − t k ) c j c k = ∫ R j ∑ c j e i t j x 2 Λ ( x ) d x ≥ 0. (15) The inequality is strict for distinct t j t_j t j
Proof. The first transform follows from (10) and Lemma 2.4; equality on the full half-plane follows by holomorphic continuation. For real ∣ z ∣ < γ 1 |z|<\gamma_1 ∣ z ∣ < γ 1
∫ R e − z x Λ ( x ) d x = c 0 − 1 E e z 2 S 1 = 1 c 0 h ( − z 2 ) . \int_{\mathbb{R}}e^{-zx}\Lambda(x)\,\mathrm{d}x=c_0^{-1}\mathbb{E}e^{z^2S_1}=\frac{1}{c_0h(-z^2)}. ∫ R e − z x Λ ( x ) d x = c 0 − 1 E e z 2 S 1 = c 0 h ( − z 2 ) 1 . The exponential moments of the resulting symmetric variable give holomorphy of the bilateral transform throughout the strip, so the identity theorem proves (14) there. Take z = i t z=it z = i t 1 / ξ ( 1 2 + t ) 1/\xi(\frac{1}{2}+t) 1/ ξ ( 2 1 + t ) (15) . Strictness follows because Λ > 0 \Lambda>0 Λ > 0
Consequently the exact inverse Fourier transform specified in Gröchenig [11 ] is
1 2 π ∫ R e − i t x ξ ( 1 2 + t ) d t = Λ ( x ) > 0. \frac{1}{2\pi}\int_{\mathbb{R}}\frac{e^{-itx}}{\xi\left(\frac{1}{2}+t\right)}\,dt=\Lambda(x)>0. 2 π 1 ∫ R ξ ( 2 1 + t ) e − i t x d t = Λ ( x ) > 0. The Gaussian mixture in (13) gives a positive representation of this inverse transform and proves the positive-definiteness property discussed after Theorem 5 of Gröchenig [11 ] . Konstantopoulos et al. [15 ] ask whether F F F D P \mathcal{D}_{P} D P F F F [15 ] derive for members of D P \mathcal{D}_{P} D P log F ( i t ) = B ( t 2 ) \log F(it)=B(t^{2}) log F ( i t ) = B ( t 2 ) ℓ \ell ℓ B B B ( t ℓ ( t ) ) ′ = H ′ ( t ) < 0 (t\ell(t))'=H'(t)<0 ( t ℓ ( t ) ) ′ = H ′ ( t ) < 0 B B B B J \mathcal{B}_{J} B J F F F D P \mathcal{D}_{P} D P [24 ] , and Polson [24 ] further restricts the possible representations.
Proof of Theorem 1.2: Log-concavity and order ten
This section proves Theorem 1.2, which has two parts with different inputs. Log-concavity needs only the low zeros (§ 3.1 ). Order ten for Λ \Lambda Λ § 3.2 , with the estimates in Appendix § 10 ).
Log-concavity from the low verified zeros
The one-sided density has a boundary at zero, where a small additive error need not preserve logarithmic curvature. Exponential tilting avoids that difficulty. A second low real zero keeps the tilted residual self-decomposable; its unimodality and its shape before the mode then control the required primitive on the whole positive axis. The same tilting argument also gives bilateral log-concavity. Only the zeros through height 100 are used in this subsection.
Log-concavity is additional information: the self-decomposable law Gamma( α , b ) (\alpha,b) ( α , b ) 0 < α < 1 0<\alpha<1 0 < α < 1 b > 0 b>0 b > 0 g ( x ) = C x α − 1 e − b x g(x)=Cx^{\alpha-1}e^{-bx} g ( x ) = C x α − 1 e − b x ( log g ) ′ ′ ( x ) = ( 1 − α ) / x 2 > 0 (\log g)''(x)=(1-\alpha)/x^{2}>0 ( log g ) ′′ ( x ) = ( 1 − α ) / x 2 > 0 S 1 S_{1} S 1 X 1 , 1 X_{1,1} X 1 , 1 Φ \Phi Φ P F 2 \mathrm{PF}_{2} PF 2 [14 ] .
We use two facts from the theory of self-decomposable laws. Every such law on R \mathbb{R} R [35 ] ). A drift-free positive self-decomposable law with a smooth positive density has Yamazato’s property: its density is log-concave before its mode and nonincreasing after it. This is Yamazato’s result, as recalled by Watanabe [33 ] , whose Theorem 1.2 extends it to unimodal one-sided Lévy processes without drift. For a variable V V V r r r e r x Pr ( V ∈ d x ) / E e r V e^{rx}\Pr(V\in dx)/\mathbb{E}e^{rV} e r x Pr ( V ∈ d x ) / E e r V k ( t ) / t k(t)/t k ( t ) / t k ( t ) k(t) k ( t ) e r t k ( t ) e^{rt}k(t) e r t k ( t )
The next elementary lemma is a localized strict version of the classical passage from a log-concave density to its distribution function; compare Bagnoli and Bergstrom [1 ] . They credit Prékopa for the integration principle and Dierker for the elementary proof. We give the precise version needed here.
Lemma 3.1 (Log-concavity of a primitive). Let g > 0 g>0 g > 0 C 1 C^{1} C 1 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) g ′ ( x ) > 0 g'(x)>0 g ′ ( x ) > 0 log g \log g log g ( 0 , x ) (0,x) ( 0 , x ) G ( x ) = ∫ 0 x g ( u ) d u G(x)=\int_{0}^{x}g(u)\,du G ( x ) = ∫ 0 x g ( u ) d u ( log G ) ′ ′ ( x ) < 0 (\log G)''(x)<0 ( log G ) ′′ ( x ) < 0 x > 0 x>0 x > 0
Proof. If g ′ ( x ) ≤ 0 g'(x) \le0 g ′ ( x ) ≤ 0 G ( x ) g ′ ( x ) − g ( x ) 2 < 0 G(x)g'(x)-g(x)^2 < 0 G ( x ) g ′ ( x ) − g ( x ) 2 < 0 r = g ′ ( x ) / g ( x ) > 0 r=g'(x)/g(x)>0 r = g ′ ( x ) / g ( x ) > 0 log g \log g log g
G ( x ) ≤ g ( x ) ∫ 0 x e r ( u − x ) d u = g ( x ) 1 − e − r x r < g ( x ) r . G(x) \le g(x)\int_0^x e^{r(u-x)}\,\mathrm{d}u = g(x)\frac{1-e^{-rx}}{r}<\frac{g(x)}{r}. G ( x ) ≤ g ( x ) ∫ 0 x e r ( u − x ) d u = g ( x ) r 1 − e − r x < r g ( x ) . Again G g ′ − g 2 < 0 Gg'-g^2<0 G g ′ − g 2 < 0
Proposition 3.2 (The one-sided reciprocal density). The one-sided density satisfies
( log Θ ) ′ ′ ( x ) < 0 ( x > 0 ) . (\log\Theta)''(x)<0 \qquad(x>0). ( log Θ ) ′′ ( x ) < 0 ( x > 0 ) . Its extension by zero to ( − ∞ , 0 ] (-\infty,0] ( − ∞ , 0 ] P F 2 \mathrm{PF}_2 PF 2
Proof. Put q = γ 1 2 ∈ ( 196 , 225 ) q=\gamma_1^2\in(196,225) q = γ 1 2 ∈ ( 196 , 225 ) m 1 m_1 m 1 η ∈ ( 21 , 22 ) \eta\in(21,22) η ∈ ( 21 , 22 ) § 10 .6. We claim that
J ( t ) = e q t ( H ( t ) − m 1 e − q t ) > 0 , J ′ ( t ) < 0 ( t > 0 ) . (16) J(t)=e^{qt}\left(H(t)-m_1e^{-qt}\right)>0,\qquad J'(t)<0 \qquad(t>0). \tag*{(16)} J ( t ) = e q t ( H ( t ) − m 1 e − q t ) > 0 , J ′ ( t ) < 0 ( t > 0 ) . (16) To see this, retain the term at η \eta η e q t H > 100 ( t ) e^{qt}H_{>100}(t) e q t H > 100 ( t ) τ = η 2 − q > 216 \tau=\eta^2-q>216 τ = η 2 − q > 216 λ ρ − q \lambda_\rho-q λ ρ − q ∣ b ρ ∣ t ≤ π / 2 |b_\rho|t\le\pi/2 ∣ b ρ ∣ t ≤ π /2 t > π / ( 2 γ ) t>\pi/(2\gamma) t > π / ( 2 γ )
( a ρ − q ) − τ = a ρ − η 2 ≥ 3 γ 2 / 4 , ∣ λ ρ − q ∣ ≤ 2 γ 2 ≤ τ γ 2 . (a_\rho-q)-\tau=a_\rho-\eta^2\ge3\gamma^2/4,\qquad|\lambda_\rho-q|\le2\gamma^2\le\tau\gamma^2. ( a ρ − q ) − τ = a ρ − η 2 ≥ 3 γ 2 /4 , ∣ λ ρ − q ∣ ≤ 2 γ 2 ≤ τ γ 2 . The proof of Lemma 2.2 now applies verbatim, giving
e − τ t + e q t H > 100 ( t ) ≥ ( 1 − ε 100 ) e − τ t , − d d t ( e − τ t + e q t H > 100 ( t ) ) ≥ ( 1 − ε 100 ) τ e − τ t . e^{-\tau t}+e^{qt}H_{>100}(t)\ge(1-\varepsilon_{100})e^{-\tau t},\qquad
-\frac{\mathrm{d}}{\mathrm{d}t}\left(e^{-\tau t}+e^{qt}H_{>100}(t)\right)\ge(1-\varepsilon_{100})\tau e^{-\tau t}. e − τ t + e q t H > 100 ( t ) ≥ ( 1 − ε 100 ) e − τ t , − d t d ( e − τ t + e q t H > 100 ( t ) ) ≥ ( 1 − ε 100 ) τ e − τ t . The other terms in J J J (16) .
Subtracting all m 1 m_1 m 1 S 1 = d Gamma ( m 1 , q ) + V S_1\mathrel{\mathop=^{\mathrm{d}}}\operatorname{Gamma}(m_1,q)+V S 1 = d Gamma ( m 1 , q ) + V V V V ( H ( t ) − m 1 e − q t ) / t (H(t)-m_1e^{-qt})/t ( H ( t ) − m 1 e − q t ) / t q q q M : = E e q V < ∞ M:=\mathbb{E}e^{qV}<\infty M := E e q V < ∞ V ( q ) V^{(q)} V ( q ) J ( t ) / t J(t)/t J ( t ) / t § 4 gives it a smooth positive density: removing finitely many factors and tilting preserve the small-time divergence of J J J
Let F q F_q F q Lemma 3.1 give ( log F q ) ′ ′ < 0 (\log F_q)''<0 ( log F q ) ′′ < 0 Exp ( q ) + V \operatorname{Exp}(q)+V Exp ( q ) + V
f ( x ) = q M e − q x F q ( x ) , x > 0 , f(x)=qMe^{-qx}F_q(x),\qquad x>0, f ( x ) = q M e − q x F q ( x ) , x > 0 , so it has the same strict logarithmic curvature. If m 1 > 1 m_1>1 m 1 > 1 m 1 − 1 m_1-1 m 1 − 1 f f f q e − q x ∫ 0 x e q u f ( u ) d u qe^{-qx}\int_0^x e^{qu}f(u)\,\mathrm{d}u q e − q x ∫ 0 x e q u f ( u ) d u e q u f ( u ) e^{qu}f(u) e q u f ( u ) Lemma 3.1 preserves strict curvature. This proves the claim for θ 1 = c 0 Θ \theta_1=c_0\Theta θ 1 = c 0 Θ P F 2 \mathrm{PF}_2 PF 2
Proposition 3.3 (The bilateral reciprocal density). The density Λ \Lambda Λ is strictly log-concave on R \mathbb{R} R Every P F 2 \mathrm{PF}_2 PF 2 minor is strictly positive.
Proof. Gaussian subordination (Sato, [27 ] , Theorem 30.1) gives the Lévy density of X 1 , 1 X_{1,1} X 1 , 1 A ( ∣ x ∣ ) / ∣ x ∣ A(|x|)/|x| A ( ∣ x ∣ ) /∣ x ∣ [24 ] , v24, Section 2 and Proposition 12.1), where
A ( x ) = ∑ ρ ∈ Z + e − γ ρ x cos ( δ ρ x ) , x > 0. A(x)=\sum_{\rho\in\mathbb{Z}_+}e^{-\gamma_\rho x}\cos(\delta_\rho x),\qquad x>0. A ( x ) = ρ ∈ Z + ∑ e − γ ρ x cos ( δ ρ x ) , x > 0. Indeed, the Gaussian integral applied to each term of H H H e − ( γ − i δ ρ ) ∣ x ∣ / ∣ x ∣ e^{-(\gamma-i\delta_\rho)|x|}/|x| e − ( γ − i δ ρ ) ∣ x ∣ /∣ x ∣ x ≠ 0 x\ne0 x = 0 ∣ x ∣ − 1 ∑ ρ e − a ρ ∣ x ∣ < ∞ |x|^{-1}\sum_\rho e^{-\sqrt{a_\rho}|x|}<\infty ∣ x ∣ − 1 ∑ ρ e − a ρ ∣ x ∣ < ∞ a = γ 1 < 15 a=\gamma_1<15 a = γ 1 < 15 ∣ s ∣ < a |s|<a ∣ s ∣ < a e s x A ( x ) e^{sx}A(x) e s x A ( x ) e − s x A ( x ) e^{-sx}A(x) e − s x A ( x ) η ∈ ( 21 , 22 ) \eta\in(21,22) η ∈ ( 21 , 22 ) x ≤ π x \le\pi x ≤ π ∣ δ ρ ∣ ≤ 1 / 2 |\delta_\rho| \le1/2 ∣ δ ρ ∣ ≤ 1/2 x > π x > \pi x > π
e − ( γ − η ) x ≤ e − π ( γ − 22 ) , ∣ γ ∓ s − i δ ρ ∣ ≤ 2 γ ≤ ( η ∓ s ) γ 2 . e^{-(\gamma-\eta)x} \le e^{-\pi(\gamma-22)}, \qquad|\gamma\mp s-i\delta_\rho| \le2\gamma\le(\eta\mp s)\gamma^2. e − ( γ − η ) x ≤ e − π ( γ − 22 ) , ∣ γ ∓ s − i δ ρ ∣ ≤ 2 γ ≤ ( η ∓ s ) γ 2 . The reciprocal-square tail bound makes the negative tail contributions in either estimate, relative to the retained term, at most
ε = 1004 e − 78 π < 1. \varepsilon= 1004e^{-78\pi} < 1. ε = 1004 e − 78 π < 1. All remaining verified terms have positive coefficients and rates γ ∓ s > 0 \gamma\mp s > 0 γ ∓ s > 0 E e s X 1 , 1 = E e s 2 S 1 < ∞ \mathbb{E}e^{sX_{1,1}}=\mathbb{E}e^{s^2S_1}<\infty E e s X 1 , 1 = E e s 2 S 1 < ∞ ∣ s ∣ < a |s|<a ∣ s ∣ < a e s x A ( ∣ x ∣ ) / ∣ x ∣ e^{sx}A(|x|)/|x| e s x A ( ∣ x ∣ ) /∣ x ∣ [27 Theorem 15.10] ), and hence unimodal.
Choose η \eta η Theorem C.3 . This extracts a Lap ( a ) \operatorname{Lap}(a) Lap ( a ) p = c 0 Λ p=c_0\Lambda p = c 0 Λ ∣ p ′ ( x ) ∣ < a p ( x ) |p'(x)|<ap(x) ∣ p ′ ( x ) ∣ < a p ( x ) x x x x x x ( log p ) ′ ′ ( x 0 ) > 0 (\log p)''(x_0)>0 ( log p ) ′′ ( x 0 ) > 0 s = − p ′ ( x 0 ) / p ( x 0 ) ∈ ( − a , a ) s=-p'(x_0)/p(x_0)\in(-a,a) s = − p ′ ( x 0 ) / p ( x 0 ) ∈ ( − a , a ) x 0 x_0 x 0 log p \log p log p p ( x ) = C e b x p(x)=Ce^{bx} p ( x ) = C e b x P F 2 \mathrm{PF}_2 PF 2
These arguments establish order two without the high verified height. They do not by themselves give order three. For example, if 0 < α < 1 0<\alpha<1 0 < α < 1 b > q b>q b > q q q q Gamma ( α , b ) \operatorname{Gamma}(\alpha,b) Gamma ( α , b ) f f f Exp ( q ) + Gamma ( α , b ) \operatorname{Exp}(q)+\operatorname{Gamma}(\alpha,b) Exp ( q ) + Gamma ( α , b ) f ( x ) = C x α ( 1 + O ( x ) ) f(x)=Cx^\alpha(1+O(x)) f ( x ) = C x α ( 1 + O ( x ))
− det [ f ( i + j ) ( x ) ] i , j = 0 2 ∼ 2 C 3 α 2 ( α − 1 ) x 3 α − 6 < 0. -\det\left[f^{(i+j)}(x)\right]_{i,j=0}^{2}\sim2C^3\alpha^2(\alpha-1)x^{3\alpha-6}<0. − det [ f ( i + j ) ( x ) ] i , j = 0 2 ∼ 2 C 3 α 2 ( α − 1 ) x 3 α − 6 < 0. Thus this density fails P F 3 \mathrm{PF}_3 PF 3 Λ \Lambda Λ § 3.2 supplies it from the verified height; Appendix § 10 .§ 5 treats each fixed order of Θ \Theta Θ
Order ten
For order ten we retain a zero near the verified height, so that the residual law is very small, and compare Λ \Lambda Λ
Proof of Theorem 1.2 . Propositions 3.2 and 3.3 prove the log-concavity assertions. For order ten, extract eleven verified Laplace factors and a symmetric residual Y U Y_U Y U U = T ∗ U=T_* U = T ∗ E Y U 2 \mathbb{E}Y_U^2 E Y U 2 E cosh ( 10 5 Y U ) − 1 \mathbb{E}\cosh(10^5Y_U)-1 E cosh ( 1 0 5 Y U ) − 1 § 10 .7, makes the signed derivative determinants positive through size ten. Lemma C.6 converts these signs into strictly positive P F 10 \mathrm{PF}_{10} PF 10 § 10 .§ 4 . □
The order-ten result also has an operator interpretation. Let ν ( f ) \nu(f) ν ( f ) f f f f f f [12 ] ) gives
ν ( Λ ∗ f ) ≤ ν ( f ) if ν ( f ) ≤ 9. \nu(\Lambda* f)\le\nu(f)\qquad\text{if }\nu(f)\le9. ν ( Λ ∗ f ) ≤ ν ( f ) if ν ( f ) ≤ 9. The convolution converges absolutely because Λ \Lambda Λ [11 ] .
Proof of Theorem 1.3: Laguerre inequalities on an interval
This section proves Theorem 1.3 by expressing each L n L_n L n
Coefficients of the squared modulus
The definition (5) also applies to real C 2 n C^{2n} C 2 n
L 2 [ f ] = 1 4 ( f ′ ′ ) 2 − 1 3 f ′ f ′ ′ ′ + 1 12 f f ( 4 ) . L_2[f] = \frac{1}{4}(f'')^2 - \frac{1}{3}f'f''' + \frac{1}{12}ff^{(4)}. L 2 [ f ] = 4 1 ( f ′′ ) 2 − 3 1 f ′ f ′′′ + 12 1 f f ( 4 ) . Lemma 4.1 (Generating and multiplication identities). For a real entire f f f x , y x,y x , y
∣ f ( x + i y ) ∣ 2 = f ( x + i y ) f ( x − i y ) = ∑ n = 0 ∞ L n [ f ] ( x ) y 2 n . |f(x+iy)|^2=f(x+iy)f(x-iy)=\sum_{n=0}^{\infty}L_n[f](x)y^{2n}. ∣ f ( x + i y ) ∣ 2 = f ( x + i y ) f ( x − i y ) = n = 0 ∑ ∞ L n [ f ] ( x ) y 2 n . For real entire f , g f,g f , g
L n [ f g ] ( x ) = ∑ k = 0 n L k [ f ] ( x ) L n − k [ g ] ( x ) . (17) L_n[fg](x)=\sum_{k=0}^{n}L_k[f](x)L_{n-k}[g](x). \tag*{(17)} L n [ f g ] ( x ) = k = 0 ∑ n L k [ f ] ( x ) L n − k [ g ] ( x ) . (17) Local uniform convergence of real entire functions f j → f f_j\to f f j → f L n [ f j ] → L n [ f ] L_n[f_j]\to L_n[f] L n [ f j ] → L n [ f ] n n n
Proof. Multiply the two Taylor series at x x x y y y y 2 n y^{2n} y 2 n ∑ j = 0 2 n i j ( − i ) 2 n − j f ( j ) ( x ) f ( 2 n − j ) ( x ) / ( j ! ( 2 n − j ) ! ) \sum_{j=0}^{2n}i^j(-i)^{2n-j}f^{(j)}(x)f^{(2n-j)}(x)/(j!(2n-j)!) ∑ j = 0 2 n i j ( − i ) 2 n − j f ( j ) ( x ) f ( 2 n − j ) ( x ) / ( j ! ( 2 n − j )!) i j ( − i ) 2 n − j = ( − 1 ) j + n i^j(-i)^{2n-j}=(-1)^{j+n} i j ( − i ) 2 n − j = ( − 1 ) j + n (5) . Absolute convergence on compact sets justifies the multiplication. Multiplying the generating identities for f f f g g g (17) . Cauchy’s integral formula implies convergence of each fixed derivative under local uniform convergence; the last statement follows from the finite sum in (5) . □
The zero-exclusion principle
The Ξ \Xi Ξ Lemma 2.1 .
Lemma 4.2 (All-order zero-exclusion principle). Let f ≢ 0 f\not\equiv0 f ≡ 0
f ( z ) = C z d e − a 0 z 2 + b 0 z ∏ ζ ∈ Z f ∗ E 1 ( z / ζ ) , E 1 ( w ) = ( 1 − w ) e w , (18) f(z)=Cz^d e^{-a_0z^2+b_0z}\prod_{\zeta\in Z_f^*}E_1(z/\zeta),\qquad E_1(w)=(1-w)e^w, \tag*{(18)} f ( z ) = C z d e − a 0 z 2 + b 0 z ζ ∈ Z f ∗ ∏ E 1 ( z / ζ ) , E 1 ( w ) = ( 1 − w ) e w , (18) where C ∈ R ∖ { 0 } C\in\mathbb{R}\setminus\{0\} C ∈ R ∖ { 0 } d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 a 0 ≥ 0 a_0\geq0 a 0 ≥ 0 b 0 ∈ R b_0\in\mathbb{R} b 0 ∈ R Z f ∗ Z_f^* Z f ∗ ∑ ζ ∈ Z f ∗ ∣ ζ ∣ − 2 < ∞ \sum_{\zeta\in Z_f^*}|\zeta|^{-2}<\infty ∑ ζ ∈ Z f ∗ ∣ ζ ∣ − 2 < ∞ I ⊂ R I\subset\mathbb{R} I ⊂ R
∣ x − Re ζ ∣ ≥ ∣ Im ζ ∣ ( x ∈ I , ζ ∈ Z f ∗ ∖ R ) . |x-\operatorname{Re}\zeta|\geq|\operatorname{Im}\zeta|\qquad(x\in I,\ \zeta\in Z_f^*\setminus\mathbb{R}). ∣ x − Re ζ ∣ ≥ ∣ Im ζ ∣ ( x ∈ I , ζ ∈ Z f ∗ ∖ R ) . Then L n [ f ] ( x ) ≥ 0 L_n[f](x)\geq0 L n [ f ] ( x ) ≥ 0 n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0 x ∈ I x\in I x ∈ I
In particular, if every nonreal zero ζ = a + i b \zeta=a+ib ζ = a + ib ∣ a ∣ > T |a|>T ∣ a ∣ > T ∣ b ∣ ≤ Δ |b|\leq\Delta ∣ b ∣ ≤ Δ T > Δ ≥ 0 T>\Delta\geq0 T > Δ ≥ 0
L n [ f ] ( x ) ≥ 0 ( n ∈ N 0 , ∣ x ∣ ≤ T − Δ ) . L_n[f](x)\geq0\qquad(n\in\mathbb{N}_0,\ |x|\leq T-\Delta). L n [ f ] ( x ) ≥ 0 ( n ∈ N 0 , ∣ x ∣ ≤ T − Δ ) . Likewise, if ∣ Im ζ ∣ ≤ Δ |\operatorname{Im}\zeta|\leq\Delta ∣ Im ζ ∣ ≤ Δ [ x − , x + ] [x_-,x_+] [ x − , x + ] x + − x − > 2 Δ x_+-x_->2\Delta x + − x − > 2Δ [ x − + Δ , x + − Δ ] [x_-+\Delta,x_+-\Delta] [ x − + Δ , x + − Δ ]
Proof. Fix x ∈ I x\in I x ∈ I Y = y 2 Y=y^2 Y = y 2 r ≠ 0 r\neq0 r = 0
E 1 ( ( x + i y ) / r ) E 1 ( ( x − i y ) / r ) = e 2 x / r r 2 ( ( x − r ) 2 + Y ) , E_1((x+iy)/r)E_1((x-iy)/r)=\frac{e^{2x/r}}{r^2}\bigl((x-r)^2+Y\bigr), E 1 (( x + i y ) / r ) E 1 (( x − i y ) / r ) = r 2 e 2 x / r ( ( x − r ) 2 + Y ) , which has nonnegative coefficients in Y Y Y z d z^d z d ( x 2 + Y ) d (x^2+Y)^d ( x 2 + Y ) d
For a conjugate pair ζ = a + i b \zeta=a+ib ζ = a + ib ζ ‾ = a − i b \overline{\zeta}=a-ib ζ = a − ib b ≠ 0 b\neq0 b = 0 Q a , b ( z ) = ( z − a ) 2 + b 2 Q_{a,b}(z)=(z-a)^2+b^2 Q a , b ( z ) = ( z − a ) 2 + b 2
E 1 ( z / ζ ) E 1 ( z / ζ ‾ ) = Q a , b ( z ) a 2 + b 2 exp ( 2 a z a 2 + b 2 ) . E_1(z/\zeta)E_1(z/\overline{\zeta})=\frac{Q_{a,b}(z)}{a^2+b^2}\exp\left(\frac{2az}{a^2+b^2}\right). E 1 ( z / ζ ) E 1 ( z / ζ ) = a 2 + b 2 Q a , b ( z ) exp ( a 2 + b 2 2 a z ) . With u = x − a u=x-a u = x − a
Q a , b ( x + i y ) Q a , b ( x − i y ) = ( u 2 + b 2 ) 2 + 2 ( u 2 − b 2 ) Y + Y 2 . Q_{a,b}(x+iy)Q_{a,b}(x-iy)=(u^2+b^2)^2+2(u^2-b^2)Y+Y^2. Q a , b ( x + i y ) Q a , b ( x − i y ) = ( u 2 + b 2 ) 2 + 2 ( u 2 − b 2 ) Y + Y 2 . Its coefficients are nonnegative by ∣ u ∣ ≥ ∣ b ∣ |u| \ge|b| ∣ u ∣ ≥ ∣ b ∣ Y Y Y e 4 a x / ( a 2 + b 2 ) / ( a 2 + b 2 ) 2 e^{4ax/(a^2+b^2)}/(a^2+b^2)^2 e 4 a x / ( a 2 + b 2 ) / ( a 2 + b 2 ) 2
Finally, the leading factor in (18) contributes
C 2 e − 2 a 0 x 2 + 2 b 0 x e 2 a 0 Y , C^2 e^{-2a_0x^2+2b_0x}e^{2a_0Y}, C 2 e − 2 a 0 x 2 + 2 b 0 x e 2 a 0 Y , whose series in Y Y Y a 0 ≥ 0 a_0 \ge0 a 0 ≥ 0 Y Y Y x + i y x+iy x + i y x − i y x-iy x − i y y y y f ( x + i y ) f ( x − i y ) f(x+iy)f(x-iy) f ( x + i y ) f ( x − i y ) L n [ f ] ( x ) L_n[f](x) L n [ f ] ( x ) f ( x ) f(x) f ( x )
For ∣ x ∣ ≤ T − Δ |x| \le T-\Delta ∣ x ∣ ≤ T − Δ ∣ x − a ∣ ≥ ∣ a ∣ − ∣ x ∣ > Δ ≥ ∣ b ∣ |x-a| \ge|a|-|x| > \Delta\ge|b| ∣ x − a ∣ ≥ ∣ a ∣ − ∣ x ∣ > Δ ≥ ∣ b ∣ a < x − a < x_- a < x − a > x + a > x_+ a > x + x ∈ [ x − + Δ , x + − Δ ] x \in[x_-+\Delta,x_+-\Delta] x ∈ [ x − + Δ , x + − Δ ] ∣ x − a ∣ > Δ ≥ ∣ b ∣ |x-a| > \Delta\ge|b| ∣ x − a ∣ > Δ ≥ ∣ b ∣
Proof of Theorem 1.3 . A zeta zero ρ = β + i γ \rho=\beta+i\gamma ρ = β + iγ z ρ = γ − i ( β − 1 2 ) z_\rho=\gamma-i(\beta-\frac{1}{2}) z ρ = γ − i ( β − 2 1 ) ∣ Im z ρ ∣ < 1 / 2 |\operatorname{Im}z_\rho|<1/2 ∣ Im z ρ ∣ < 1/2 ∣ Re z ρ ∣ > T |\operatorname{Re}z_\rho|>T ∣ Re z ρ ∣ > T T T T Δ = 1 / 2 \Delta=1/2 Δ = 1/2 c 0 ∏ γ ≤ Y ( 1 − z 2 / λ ρ ) c_0\prod_{\gamma\le Y}(1-z^2/\lambda_\rho) c 0 ∏ γ ≤ Y ( 1 − z 2 / λ ρ )
To obtain strictness, fix x x x Ξ ( z ) = ( z − x ) m x g ( z ) \Xi(z)=(z-x)^{m_x}g(z) Ξ ( z ) = ( z − x ) m x g ( z ) g ( x ) ≠ 0 g(x)\ne0 g ( x ) = 0 ∣ x − Re ζ ∣ > ∣ Im ζ ∣ |x-\operatorname{Re}\zeta|>|\operatorname{Im}\zeta| ∣ x − Re ζ ∣ > ∣ Im ζ ∣ Y Y Y k ≥ 0 k\ge0 k ≥ 0 P P P Y Y Y k k k k k k g / P g/P g / P x x x L k [ g ] ( x ) ≥ L k [ P ] ( x ) ( g / P ) ( x ) 2 > 0 L_k[g](x)\ge L_k[P](x)(g/P)(x)^2>0 L k [ g ] ( x ) ≥ L k [ P ] ( x ) ( g / P ) ( x ) 2 > 0 ∣ Ξ ( x + i y ) ∣ 2 = y 2 m x ∣ g ( x + i y ) ∣ 2 |\Xi(x+iy)|^2=y^{2m_x}|g(x+iy)|^2 ∣Ξ ( x + i y ) ∣ 2 = y 2 m x ∣ g ( x + i y ) ∣ 2 F F F
Related work
The reciprocal law
The reciprocal law was asked for, and constructed under RH, by Roynette and Yor [26 ] (eq. (0.29) and Théorème V.3.2). Polson’s work connects reciprocal entire functions, Hadamard products, and Thorin representations (Polson [22 ] , § 2.1 ). In Polson [24 ] , v24, Theorems A and 5.3, whose T T T S 1 S_1 S 1 H ( t ) / t H(t)/t H ( t ) / t [23 ] , v9, Theorem 22 also gives the self-decomposable law. The two proofs use the explicit formula near zero and, at later times, verification through heights 1000 and 200, respectively. Our proof extends their large-time comparison with one low zero to every t > 0 t>0 t > 0 S α S_\alpha S α 2 S 1 S_1 S 1 [24 ] , v24, Corollary 6.8(iv); Polson [23 ] , v9, Theorem 22(ii)), which Theorem 1.2 strengthens to log-concavity.
Pólya frequency orders
Gröchenig’s Xi specialization of Schoenberg’s transform theorem (Schoenberg, [30 ] ; Gröchenig, [11 ] ) identifies Θ , Λ ∈ P F ∞ \Theta,\Lambda\in\mathrm{PF}_\infty Θ , Λ ∈ PF ∞ Λ \Lambda Λ [26 ] , Théorème V.3.2); see also Konstantopoulos et al. [15 ] , Corollary 2.4). Theorem 1.2 establishes finite PF orders for these densities. Its log-concavity argument combines Yamazato’s one-sided shape property (Yamazato, [35 ] ; Watanabe, [33 ] ) with estimates for exponential tilts; the higher-order argument applies the classical Wronskian criterion and composition formula [13 , 12 ] to a finite convolution and its residual law.
Laguerre inequalities
Csordas et al. [8 ] prove the Laguerre inequality on intervals away from the nonreal zeros. Let f ( z ) = C e − a z 2 + b z ∏ ζ ( 1 − z / ζ ) e z / ζ f(z)=Ce^{-az^{2}+bz}\prod_{\zeta}(1-z/\zeta)e^{z/\zeta} f ( z ) = C e − a z 2 + b z ∏ ζ ( 1 − z / ζ ) e z / ζ C ≠ 0 C\ne0 C = 0 a ≥ 0 a\ge0 a ≥ 0 ∣ Im z ∣ < 1 |\operatorname{Im}z|<1 ∣ Im z ∣ < 1 [ x − , x + ] [x_{-},x_{+}] [ x − , x + ] x + − x − > 2 x_{+}-x_{-}>2 x + − x − > 2 L 1 [ f ] > 0 L_{1}[f]>0 L 1 [ f ] > 0 [ x − + 1 , x + − 1 ] [x_{-}+1,x_{+}-1] [ x − + 1 , x + − 1 ] [6 ] . Their proof uses the second logarithmic derivative. Our coefficient argument is the classical product argument for Csordas and Varga [7 ] , applied to conjugate zero pairs; it gives every L n L_{n} L n Ξ \Xi Ξ
Equivalent forms of RH
The equivalence of RH with complete monotonicity of H H H [36 ] . Polson’s RH equivalences (Polson, 2026b, v24, Proposition 3.2 and Theorem 5.6; Polson, 2026a, v9, Theorem 26) identify complete monotonicity of H H H S 1 S_{1} S 1 H / t α H/t^{\alpha} H / t α α > 0 \alpha>0 α > 0 [24 ] shows moreover that Hankel positivity of every order at a single time, with both Hankel blocks of the Stieltjes moment problem, is equivalent to RH; Polson [25 ] discusses the unshifted test.
The xi function in probability
The xi function itself also occurs as a probability transform. Biane et al. [3 ] survey the probability laws whose transforms are ξ \xi ξ [17 ] shows that ξ ( σ − i t ) / ξ ( σ ) \xi(\sigma-it)/\xi(\sigma) ξ ( σ − i t ) / ξ ( σ ) t t t σ \sigma σ pretended infinite divisibility for every 1 / 2 < σ < 1 1/2<\sigma<1 1/2 < σ < 1 [16 ] retains that condition. Nakamura and Suzuki [18 ] construct from the explicit formula a function that is the characteristic function of an infinitely divisible law if and only if RH holds. Under RH its Lévy measure is ∑ γ m γ γ − 2 δ − γ \sum_{\gamma}m_{\gamma}\gamma^{-2}\delta_{-\gamma} ∑ γ m γ γ − 2 δ − γ γ \gamma γ m γ m_{\gamma} m γ 1 / h 1/h 1/ h H ( t ) / t H(t)/t H ( t ) / t ( F ( t ) , 1 / F ( i t ) ) (F(t),1/F(it)) ( F ( t ) , 1/ F ( i t )) [15 ] .
AI usage
The results in this paper were developed through extensive use of an AI assistant (OpenAI’s GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The reciprocal-density revisions, verification programs, and attribution review also used OpenAI Codex. Anthropic’s Claude (Opps 5.5 and Fable 5.1) was used extensively to assist with the writing. Assistant checks and program reruns are not external peer review or formal verification.
Acknowledgements
The author (SS) thanks the Alexander von Humboldt (AvH) Foundation for their generous support via a Humboldt Professorship in AI.
Equivalent forms of the Riemann hypothesis
Extended to every order, the positivity properties of § 2.3 and § 3 are each equivalent to RH. Schoenberg’s characterization relates P F ∞ \mathrm{PF}_{\infty} PF ∞ [30 ] . Its Xi specialization [26 , 11 Theorems 3–5] combines with the theory of complete Bernstein functions to give the following equivalences. A Bernstein function B B B complete , written B ∈ C B F B \in\mathrm{CBF} B ∈ CBF [24 ] and Polson [23 ] ; the continuation argument goes back to Polson [21 ] .
Proposition A.1 (Equivalent forms of the Riemann hypothesis). The following assertions are equivalent:
(i) RH;
(ii) H H H
(iii) H ( t ) / t H(t)/t H ( t ) / t
(iv) for one (equivalently, every) finite α > 0 \alpha> 0 α > 0 H ( t ) / t α H(t)/t^\alpha H ( t ) / t α
(v) [ H ( u i + u j ) ] [H(u_i + u_j)] [ H ( u i + u j )] u i > 0 u_i > 0 u i > 0
(vi) B ∈ C B F B \in\mathrm{CBF} B ∈ CBF
(vii) Θ ∈ P F ∞ \Theta\in\mathrm{PF}_{\infty} Θ ∈ PF ∞
(viii) Λ ∈ P F ∞ \Lambda\in\mathrm{PF}_{\infty} Λ ∈ PF ∞
Under these equivalent conditions the Hankel kernels of H H H H / t α H/t^\alpha H / t α α > 0 \alpha> 0 α > 0
The central implication is analytic. Complete monotonicity gives a positive Laplace representation, which extends h ′ / h h'/h h ′ / h log h \log h log h h h h
Proof. We prove the implications in the order
(i) ⇒ (ii) ⇒ { (iii) , (iv) } , (ii) ⇔ (v) , (iii) ⇔ (vi) ⇒ (i) , \text{(i)} \Rightarrow\text{(ii)} \Rightarrow\{\text{(iii)},\text{ (iv)}\}, \qquad
\text{(ii)} \Leftrightarrow\text{(v)}, \qquad
\text{(iii)} \Leftrightarrow\text{(vi)} \Rightarrow\text{(i)}, (i) ⇒ (ii) ⇒ { (iii) , (iv) } , (ii) ⇔ (v) , (iii) ⇔ (vi) ⇒ (i) , (iv) ⇒ (i) , (i) ⇔ (vii) ⇔ (viii) . \text{(iv)} \Rightarrow\text{(i)}, \qquad
\text{(i)} \Leftrightarrow\text{(vii)} \Leftrightarrow\text{(viii)}. (iv) ⇒ (i) , (i) ⇔ (vii) ⇔ (viii) . Under RH, λ ρ = γ 2 > 0 \lambda_\rho= \gamma^2 > 0 λ ρ = γ 2 > 0 H H H t − α t^{-\alpha} t − α α > 0 \alpha> 0 α > 0
H ( t ) = ∫ R e − t x ν ( d x ) H(t) = \int_{\mathbb{R}} e^{-tx}\nu(dx) H ( t ) = ∫ R e − t x ν ( d x ) with ν ≥ 0 \nu\ge0 ν ≥ 0 H ( t ) → 0 H(t) \to0 H ( t ) → 0 t → ∞ t \to\infty t → ∞ ν \nu ν ( − ∞ , 0 ] (-\infty, 0] ( − ∞ , 0 ] H H H H / t H/t H / t [28 Chapter 6] makes B B B H / t H/t H / t
If B B B C ∖ ( − ∞ , 0 ] \mathbb{C} \setminus(-\infty, 0] C ∖ ( − ∞ , 0 ] h = e B h = e^B h = e B h h h h ( 0 ) = 1 h(0) = 1 h ( 0 ) = 1 ρ = β + i γ ∈ Z + \rho= \beta+ i\gamma\in\mathbb{Z}_+ ρ = β + iγ ∈ Z +
( ρ − 1 2 ) 2 = δ ρ 2 − γ 2 + 2 i γ δ ρ . \left(\rho- \frac{1}{2}\right)^2 = \delta_\rho^2 - \gamma^2 + 2i\gamma\delta_\rho. ( ρ − 2 1 ) 2 = δ ρ 2 − γ 2 + 2 iγ δ ρ . As γ > 0 \gamma> 0 γ > 0 δ ρ = 0 \delta_\rho= 0 δ ρ = 0 h h h ℓ α ( t ) = H ( t ) t − α \ell_\alpha(t)=H(t)t^{-\alpha} ℓ α ( t ) = H ( t ) t − α α > 0 \alpha>0 α > 0 ℓ α ( t ) = ∫ 0 ∞ e − r t μ ( d r ) \ell_\alpha(t)=\int_0^\infty e^{-rt}\,\mu(dr) ℓ α ( t ) = ∫ 0 ∞ e − r t μ ( d r ) μ ≥ 0 \mu\geq0 μ ≥ 0 s > 0 s>0 s > 0
B ′ ( s ) = ∫ 0 ∞ e − s t H ( t ) d t = Γ ( α + 1 ) ∫ 0 ∞ ( s + r ) − α − 1 μ ( d r ) . B'(s)=\int_0^\infty e^{-st}H(t)\,dt=\Gamma(\alpha+1)\int_0^\infty(s+r)^{-\alpha-1}\,\mu(dr). B ′ ( s ) = ∫ 0 ∞ e − s t H ( t ) d t = Γ ( α + 1 ) ∫ 0 ∞ ( s + r ) − α − 1 μ ( d r ) . At s = 1 s=1 s = 1 ∫ ( 1 + r ) − α − 1 μ ( d r ) < ∞ \int(1+r)^{-\alpha-1}\,\mu(dr)<\infty ∫ ( 1 + r ) − α − 1 μ ( d r ) < ∞ K ⊂ C ∖ ( − ∞ , 0 ] K\subset\mathbb{C}\setminus(-\infty,0] K ⊂ C ∖ ( − ∞ , 0 ] ∣ z + r ∣ ≥ c K ( 1 + r ) |z+r|\geq c_K(1+r) ∣ z + r ∣ ≥ c K ( 1 + r ) z ∈ K z\in K z ∈ K r ≥ 0 r\geq0 r ≥ 0 c K > 0 c_K>0 c K > 0 J J J [23 ] . The identity theorem gives h ′ = J h h'=Jh h ′ = J h h h h m m m m − 1 m-1 m − 1 m m m h h h ⇒ \Rightarrow ⇒
Under RH the genus-zero product (9) becomes
h ( s ) = ∏ γ > 0 ( 1 + s γ 2 ) , Ξ ( z ) = c 0 ∏ γ > 0 ( 1 − z 2 γ 2 ) , h(s)=\prod_{\gamma>0}\left(1+\frac{s}{\gamma^2}\right),\qquad\Xi(z)=c_0\prod_{\gamma>0}\left(1-\frac{z^2}{\gamma^2}\right), h ( s ) = γ > 0 ∏ ( 1 + γ 2 s ) , Ξ ( z ) = c 0 γ > 0 ∏ ( 1 − γ 2 z 2 ) , where ordinates are counted with multiplicity and ∑ γ > 0 γ − 2 < ∞ \sum_{\gamma>0}\gamma^{-2}<\infty ∑ γ > 0 γ − 2 < ∞ 1 / ( c 0 h ) 1/(c_0h) 1/ ( c 0 h ) 1 / Ξ 1/\Xi 1/Ξ [26 , 11 Theorems 3–5] ). Uniqueness of the Laplace transforms identifies those densities with (13) . Under RH the representing measures of H H H H / t α H/t^\alpha H / t α Proposition C.2 then gives strict total positivity in every finite order. □
Appendix B. Estimates for H and the reciprocal laws
The reciprocal-law construction needs control of H H H H H H Table 1 lists, for each result of the paper, the inputs beyond elementary calculus and linear algebra.
Table 1. Numerical inputs by result.
The counting constant
For Y ≥ e Y \ge e Y ≥ e [2 ] give the upper bound
0.11200 log Y + 0.12567 log log Y + 3.77417. 0.11200\log Y + 0.12567\log\log Y + 3.77417. 0.11200 log Y + 0.12567 log log Y + 3.77417. Put u = log Y ≥ 1 u=\log Y \ge1 u = log Y ≥ 1 ( log Y ) / 8 (\log Y)/8 ( log Y ) /8 0.12567 log u − 0.013 u + 3.77417 0.12567\log u - 0.013u + 3.77417 0.12567 log u − 0.013 u + 3.77417 u = 0.12567 / 0.013 u=0.12567/0.013 u = 0.12567/0.013 3.934 < 4 3.934 < 4 3.934 < 4 Y = e Y=e Y = e N ( e ) < 4 N(e)<4 N ( e ) < 4 N ( Y ) ≤ 3 N(Y)\le3 N ( Y ) ≤ 3 − 1 < M ( Y ) ≤ 0 -1<M(Y)\le0 − 1 < M ( Y ) ≤ 0 [ 0 , e ] [0,e] [ 0 , e ]
The theta normalization
The classical theta formula, in our normalization, is
Ξ ( z ) = ∫ R e i z x Φ ( x ) d x , Φ ( x ) = ∑ n ≥ 1 ( 4 π 2 n 4 e 9 x / 2 − 6 π n 2 e 5 x / 2 ) e − π n 2 e 2 x . (19) \Xi(z)=\int_{\mathbb{R}}e^{izx}\Phi(x)\,\mathrm{d}x,\qquad\Phi(x)=\sum_{n\ge1}\left(4\pi^2n^4e^{9x/2}-6\pi n^2e^{5x/2}\right)e^{-\pi n^2e^{2x}}. \tag*{(19)} Ξ ( z ) = ∫ R e i z x Φ ( x ) d x , Φ ( x ) = n ≥ 1 ∑ ( 4 π 2 n 4 e 9 x /2 − 6 π n 2 e 5 x /2 ) e − π n 2 e 2 x . (19) This normalization follows by differentiating the Jacobi theta identity and integrating by parts; see Titchmarsh [32 ] and Konstantopoulos et al. [15 ] . The kernel is even by the theta transformation, and each summand is positive for x ≥ 0 x\ge0 x ≥ 0 2 π n 2 e 2 x > 3 2\pi n^2e^{2x}>3 2 π n 2 e 2 x > 3 F F F Φ / c 0 \Phi/c_0 Φ/ c 0
Asymptotics of H at both ends.
Proposition B.1 (Asymptotics). As t ↓ 0 t\downarrow0 t ↓ 0
H ( t ) = log ( 1 / t ) − γ E − 2 log ( 4 π ) 8 π t + O ( log ( 1 / t ) ) . (20) H(t)=\frac{\log(1/t)-\gamma_E-2\log(4\pi)}{8\sqrt{\pi t}}+O(\log(1/t)). \tag*{(20)} H ( t ) = 8 π t log ( 1/ t ) − γ E − 2 log ( 4 π ) + O ( log ( 1/ t )) . (20) For each fixed nonnegative integer j j j
( − 1 ) j H ( j ) ( t ) ∼ Γ ( j + 1 2 ) 8 π t − j − 1 / 2 log ( 1 / t ) . (21) (-1)^jH^{(j)}(t)\sim\frac{\Gamma\left(j+\frac{1}{2}\right)}{8\pi}t^{-j-1/2}\log(1/t). \tag*{(21)} ( − 1 ) j H ( j ) ( t ) ∼ 8 π Γ ( j + 2 1 ) t − j − 1/2 log ( 1/ t ) . (21) At infinity H ( t ) = m 1 e − q t + O ( e − ( q + ε ) t ) H(t)=m_1e^{-qt}+O(e^{-(q+\varepsilon)t}) H ( t ) = m 1 e − q t + O ( e − ( q + ε ) t ) ε > 0 \varepsilon>0 ε > 0 m 1 m_1 m 1
The leading term in (B.2) agrees with Polson [24 ] , who also obtains the constant term 7 / 8 7/8 7/8
Proof. First replace every λ ρ \lambda_\rho λ ρ γ 2 \gamma^2 γ 2 (6) gives the main integral
1 2 π ∫ 0 ∞ e − t y 2 log ( y / ( 2 π ) ) d y . \frac{1}{2\pi}\int_0^\infty e^{-ty^2}\log(y/(2\pi))\,\mathrm{d}y. 2 π 1 ∫ 0 ∞ e − t y 2 log ( y / ( 2 π )) d y . The error is bounded by ∫ 0 ∞ 2 t y e − t y 2 [ 1 8 log ( y + 2 ) + 4 ] d y = O ( 1 + log ( 1 / t ) ) \int_0^\infty2tye^{-ty^2}\left[\frac{1}{8}\log(y+2)+4\right]\,\mathrm{d}y=O(1+\log(1/t)) ∫ 0 ∞ 2 t y e − t y 2 [ 8 1 log ( y + 2 ) + 4 ] d y = O ( 1 + log ( 1/ t )) x = t y x=\sqrt{t}\,y x = t y
∫ 0 ∞ e − x 2 d x = π 2 \int_0^\infty e^{-x^2}\,\mathrm{d}x=\frac{\sqrt{\pi}}{2} ∫ 0 ∞ e − x 2 d x = 2 π and
∫ 0 ∞ e − x 2 log x d x = π ( − γ E − 2 log 2 ) 4 \int_0^\infty e^{-x^2}\log x\,\mathrm{d}x=\frac{\sqrt{\pi}(-\gamma_E-2\log2)}{4} ∫ 0 ∞ e − x 2 log x d x = 4 π ( − γ E − 2 log 2 ) to obtain the stated main term. For 0 < t ≤ 1 0<t\le1 0 < t ≤ 1
∣ e t δ 2 cos ( 2 t γ δ ) − 1 ∣ ≤ C ( t + t 2 γ 2 ) . \left|e^{t\delta^2}\cos(2t\gamma\delta)-1\right|\le C(t+t^2\gamma^2). e t δ 2 cos ( 2 t γ δ ) − 1 ≤ C ( t + t 2 γ 2 ) . Zero counting then bounds the total replacement error by O ( t [ 1 + log ( 1 / t ) ] ) O(\sqrt{t}[1+\log(1/t)]) O ( t [ 1 + log ( 1/ t )]) y 2 j e − t y 2 y^{2j}e^{-ty^2} y 2 j e − t y 2 O ( t − j log ( 1 / t ) ) O(t^{-j}\log(1/t)) O ( t − j log ( 1/ t )) z 2 j e − t z 2 z^{2j}e^{-tz^2} z 2 j e − t z 2 γ \gamma γ γ − i δ \gamma-i\delta γ − i δ O ( t 1 / 2 − j log ( 1 / t ) ) O(t^{1/2-j}\log(1/t)) O ( t 1/2 − j log ( 1/ t )) j j j ( y 2 j − 2 + t y 2 j + t 2 y 2 j + 2 ) e − t y 2 / 2 (y^{2j-2}+ty^{2j}+t^2y^{2j+2})e^{-ty^2/2} ( y 2 j − 2 + t y 2 j + t 2 y 2 j + 2 ) e − t y 2 /2 j = 0 j=0 j = 0 N N N Γ ( j + 1 / 2 ) / ( 8 π ) \Gamma(j+1/2)/(8\pi) Γ ( j + 1/2 ) / ( 8 π ) 100 2 − 1 / 4 > q 100^2-1/4>q 10 0 2 − 1/4 > q q q q t = 1 t=1 t = 1
Regularity and moments of the reciprocal laws
The Bernstein representation gives a probability measure, but the later shape arguments need a smooth density that is positive in the interior of its support. We complete that step by extracting gamma convolution factors of arbitrarily high smoothness; the asymptotics of H H H
Proof of Lemma 2.4. For each integer N ≥ 2 N \ge2 N ≥ 2 b N > 0 b_N > 0 b N > 0 α H ( t ) ≥ N e − b N t \alpha H(t) \ge N e^{-b_N t} α H ( t ) ≥ N e − b N t t > 0 t > 0 t > 0 t N > 0 t_N > 0 t N > 0 α H ≥ N \alpha H \ge N α H ≥ N ( 0 , t N ] (0,t_N] ( 0 , t N ] b N ≥ q b_N \ge q b N ≥ q N e − ( b N − q ) t N ≤ α ( 1 − ε 100 ) N e^{-(b_N-q)t_N} \le\alpha(1-\varepsilon_{100}) N e − ( b N − q ) t N ≤ α ( 1 − ε 100 )
S α = d Gamma ( N , b N ) + R N . (22) S_\alpha\mathrel{\overset{d}{=}} \operatorname{Gamma}(N,b_N)+R_N . \tag*{(22)} S α = d Gamma ( N , b N ) + R N . (22) The gamma density, extended by zero to the negative half-line, has bounded continuous derivatives through order N − 2 N-2 N − 2 R N R_N R N N N N θ α \theta_\alpha θ α R N R_N R N s s s Y Y Y lim s → ∞ − s − 1 log E e − s Y = inf supp Y \lim_{s\to\infty}-s^{-1}\log\mathbb{E}e^{-sY}=\inf\operatorname{supp}Y lim s → ∞ − s − 1 log E e − s Y = inf supp Y inf supp R N = 0 \inf\operatorname{supp}R_N=0 inf supp R N = 0 θ α ( x ) > 0 \theta_\alpha(x)>0 θ α ( x ) > 0 x > 0 x>0 x > 0
The large-time expansion of Proposition B.1 shows ∫ 0 ∞ ( e r t − 1 ) H ( t ) d t / t < ∞ \int_0^\infty(e^{rt}-1)H(t)\,\mathrm{d}t/t<\infty ∫ 0 ∞ ( e r t − 1 ) H ( t ) d t / t < ∞ r < q r<q r < q (1) gives
B ( s ) = 1 4 s log s + O ( s ) ( s → ∞ ) . (23) B(s)=\frac{1}{4}\sqrt{s}\log s+O(\sqrt{s})\qquad(s\to\infty). \tag*{(23)} B ( s ) = 4 1 s log s + O ( s ) ( s → ∞ ) . (23) In particular h ( s ) − α → 0 h(s)^{-\alpha}\to0 h ( s ) − α → 0 p > 0 p>0 p > 0
E S α − p = 1 Γ ( p ) ∫ 0 ∞ s p − 1 h ( s ) − α d s < ∞ , \mathbb{E}S_\alpha^{-p}=\frac{1}{\Gamma(p)}\int_0^\infty s^{p-1}h(s)^{-\alpha}\,\mathrm{d}s<\infty, E S α − p = Γ ( p ) 1 ∫ 0 ∞ s p − 1 h ( s ) − α d s < ∞ , using (B.5). □
Higher Pólya frequency orders
The order-ten part of Theorem 1.2 compares Λ \Lambda Λ § 3.1 . We then bound the exponential moments of the residual and certify the derivative determinants. A separate argument treats each fixed order of Θ \Theta Θ
Wronskian transfers
The following is the classical Wronskian criterion for an extended complete Chebyshev system ([13 ] ). We include the short proof to fix the signs used in the PF arguments of this appendix and in the last assertion of Proposition A.1.
Lemma C.1. Suppose real C k − 1 C^{k-1} C k − 1 f 1 , … , f k f_1,\ldots,f_k f 1 , … , f k W ( f 1 , … , f j ) = det [ f b ( a − 1 ) ] a , b = 1 j W(f_1,\ldots,f_j)=\det[f_b^{(a-1)}]_{a,b=1}^j W ( f 1 , … , f j ) = det [ f b ( a − 1 ) ] a , b = 1 j W ( f 1 , … , f j ) > 0 W(f_1,\ldots,f_j)>0 W ( f 1 , … , f j ) > 0 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k det [ f j ( s i ) ] i , j = 1 k > 0 \det[f_j(s_i)]_{i,j=1}^k>0 det [ f j ( s i ) ] i , j = 1 k > 0 s 1 < ⋯ < s k s_1<\cdots<s_k s 1 < ⋯ < s k
Proof. Induct on k k k f 1 > 0 f_1>0 f 1 > 0 s i s_i s i f 1 ( s i ) f_1(s_i) f 1 ( s i ) g j = ( f j + 1 / f 1 ) ′ g_j=(f_{j+1}/f_1)' g j = ( f j + 1 / f 1 ) ′
W ( g 1 , … , g j ) = W ( f 1 , … , f j + 1 ) f 1 j + 1 > 0. W(g_1,\ldots,g_j)=\frac{W(f_1,\ldots,f_{j+1})}{f_1^{j+1}}>0. W ( g 1 , … , g j ) = f 1 j + 1 W ( f 1 , … , f j + 1 ) > 0. Subtract consecutive normalized rows, starting at the bottom, and expand in the first column. The remaining determinant is the integral, over [ s 1 , s 2 ] × ⋯ × [ s k − 1 , s k ] [s_1,s_2] \times\cdots\times[s_{k-1},s_k] [ s 1 , s 2 ] × ⋯ × [ s k − 1 , s k ] g 1 , … , g k − 1 g_1,\ldots,g_{k-1} g 1 , … , g k − 1
Positive differential Hankel determinants pass to distinct arguments by the same principle.
Proposition C.2. If a smooth real f f f [ ( − 1 ) i + j f ( i + j ) ( t ) ] i , j = 0 k − 1 ≻ 0 [(-1)^{i+j}f^{(i+j)}(t)]_{i,j=0}^{k-1} \succ0 [( − 1 ) i + j f ( i + j ) ( t ) ] i , j = 0 k − 1 ≻ 0 t > 0 t>0 t > 0 f ( x + y ) f(x+y) f ( x + y ) k k k
Proof. Every initial differential Hankel determinant det [ f ( i + j ) ( t ) ] i , j = 0 r − 1 \det[f^{(i+j)}(t)]_{i,j=0}^{r-1} det [ f ( i + j ) ( t ) ] i , j = 0 r − 1 y y y x ↦ f ( x + y ) , f ′ ( x + y ) , … , f ( r − 1 ) ( x + y ) x \mapsto f(x+y), f'(x+y),\ldots,f^{(r-1)}(x+y) x ↦ f ( x + y ) , f ′ ( x + y ) , … , f ( r − 1 ) ( x + y ) det [ f ( j − 1 ) ( x i + y ) ] i , j = 1 r \det[f^{(j-1)}(x_i+y)]_{i,j=1}^{r} det [ f ( j − 1 ) ( x i + y ) ] i , j = 1 r x i x_i x i y ↦ f ( x i + y ) y \mapsto f(x_i+y) y ↦ f ( x i + y )
Explicit PF convolution factors
For this factorization we use a variant of Lemma 2.2. Its bounds hold with q q q q ∗ q_* q ∗ 2 ≤ q ∗ ≤ U 2 / 4 − 1 / 4 2 \le q_* \le U^2/4 - 1/4 2 ≤ q ∗ ≤ U 2 /4 − 1/4 K U ( t ) = e − q ∗ t + H > U ( t ) K_U(t) = e^{-q_*t} + H_{>U}(t) K U ( t ) = e − q ∗ t + H > U ( t ) a ρ − q ∗ ≥ 3 γ ∗ 2 / 4 a_\rho-q_* \ge3\gamma_*^2/4 a ρ − q ∗ ≥ 3 γ ∗ 2 /4 ∣ λ ρ ∣ ≤ q ∗ γ ∗ 2 |\lambda_\rho| \le q_*\gamma_*^2 ∣ λ ρ ∣ ≤ q ∗ γ ∗ 2
For 100 ≤ U ≤ T ∗ 100 \le U \le T_* 100 ≤ U ≤ T ∗ γ ∗ \gamma_* γ ∗ 2 ≤ γ ∗ 2 ≤ U 2 / 4 − 1 / 4 2 \le\gamma_*^2 \le U^2/4 - 1/4 2 ≤ γ ∗ 2 ≤ U 2 /4 − 1/4 A U A_U A U γ ≤ U \gamma\le U γ ≤ U γ ∗ \gamma_* γ ∗
P U ( s ) = ∏ γ ∈ A U ( 1 + s / γ 2 ) . P_U(s)=\prod_{\gamma\in A_U}(1+s/\gamma^2). P U ( s ) = γ ∈ A U ∏ ( 1 + s / γ 2 ) . Repeated zeros are retained with their multiplicities.
Theorem C.3 (Extraction of verified-zero PF factors). There is a positive self-decomposable random variable R U R_U R U
S 1 = d ∑ γ ∈ A U E γ + R U , E γ ∼ Exp ( γ 2 ) , (24) S_1 \mathrel{\overset{d}{=}} \sum_{\gamma\in A_U} E_\gamma+R_U,\qquad E_\gamma\sim\operatorname{Exp}(\gamma^2), \tag*{(24)} S 1 = d γ ∈ A U ∑ E γ + R U , E γ ∼ Exp ( γ 2 ) , (24) X 1 , 1 = d ∑ γ ∈ A U L γ + Y U , L γ ∼ Lap ( γ ) , (25) X_{1,1} \mathrel{\overset{d}{=}} \sum_{\gamma\in A_U} L_\gamma+Y_U,\qquad L_\gamma\sim\operatorname{Lap}(\gamma), \tag*{(25)} X 1 , 1 = d γ ∈ A U ∑ L γ + Y U , L γ ∼ Lap ( γ ) , (25) where Lap ( γ ) \operatorname{Lap}(\gamma) Lap ( γ ) γ e − γ ∣ x ∣ / 2 \gamma e^{-\gamma|x|}/2 γ e − γ ∣ x ∣ /2 Y U = d 2 R U G 1 Y_U\mathrel{\overset{d}{=}}\sqrt{2R_U}G_1 Y U = d 2 R U G 1 G 1 G_1 G 1 R U R_U R U ∞ _\infty ∞ R U R_U R U ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Y U Y_U Y U
Under RH, 1 / h ( s ) = ∏ γ ( 1 + s / γ 2 ) − 1 1/h(s)=\prod_\gamma(1+s/\gamma^2)^{-1} 1/ h ( s ) = ∏ γ ( 1 + s / γ 2 ) − 1 [4 Chapter 3] , and correspondingly X 1 , 1 = ∑ γ L γ X_{1,1}=\sum_\gamma L_\gamma X 1 , 1 = ∑ γ L γ x i x_i x i [22 ] and Polson [23 ] . The point here is the unconditional extraction of finitely many verified factors with a self-decomposable residual.
Proof. Set C U ( s ) = B ( s ) − log P U ( s ) C_U(s)=B(s)-\log P_U(s) C U ( s ) = B ( s ) − log P U ( s )
C U ( s ) = ∫ 0 ∞ ( 1 − e − s t ) K U ( t ) t d t , E e − s R U = e − C U ( s ) = P U ( s ) h ( s ) . (26) C_U(s)=\int_0^\infty(1-e^{-st})\frac{K_U(t)}{t}\,dt,\qquad\mathbb{E}e^{-sR_U}=e^{-C_U(s)}=\frac{P_U(s)}{h(s)}. \tag*{(26)} C U ( s ) = ∫ 0 ∞ ( 1 − e − s t ) t K U ( t ) d t , E e − s R U = e − C U ( s ) = h ( s ) P U ( s ) . (26) Here K U ( t ) = e − γ ∗ 2 t + H > U ( t ) K_U(t)=e^{-\gamma_*^2t}+H_{>U}(t) K U ( t ) = e − γ ∗ 2 t + H > U ( t ) Lemma 2.2 stated above. Thus the Poisson and self-decomposability constructions from § 2.3 apply. Multiplication of Laplace transforms proves (C.1). On the Fourier axis each factor becomes ( 1 + t 2 / γ 2 ) − 1 (1+t^2/\gamma^2)^{-1} ( 1 + t 2 / γ 2 ) − 1 Lap ( γ ) \operatorname{Lap}(\gamma) Lap ( γ ) ∞ _\infty ∞ [30 ] . Equivalently, their transform reciprocals have only real zeros of the required sign. Removing finitely many exponentials leaves the small-time divergence of H H H B ( s ) B(s) B ( s ) O U ( log ( 2 + s ) ) O_U(\log(2+s)) O U ( log ( 2 + s )) § 2.3 therefore apply to both residuals. □
The densities in (13) therefore have explicit finite P F ∞ \mathrm{PF}_{\infty} PF ∞
Corollary C.4 (Approximation by verified-zero PF laws). For 1000 ≤ U ≤ T ∗ 1000 \le U \le T_\ast 1000 ≤ U ≤ T ∗ ( U / 4 , U / 3 ] (U/4,U/3] ( U /4 , U /3 ]
b U = 16 U 2 + 2 ( log U + 1 ) U , A U = ∑ γ ∈ A U E γ , Z U = ∑ γ ∈ A U L γ . b_U=\frac{16}{U^2}+\frac{2(\log U+1)}{U}, \qquad A_U=\sum_{\gamma\in A_U}E_\gamma,\qquad Z_U=\sum_{\gamma\in A_U}L_\gamma. b U = U 2 16 + U 2 ( log U + 1 ) , A U = γ ∈ A U ∑ E γ , Z U = γ ∈ A U ∑ L γ . Then E R U ≤ b U \mathbb{E}R_U\le b_U E R U ≤ b U d T V d_{\mathrm{TV}} d TV [ 0 , 1 ] [0,1] [ 0 , 1 ]
d T V ( S 1 , A U ) ≤ q b U , d T V ( X 1 , 1 , Z U ) ≤ q b U , q = γ 1 2 < 225. (27) d_{\mathrm{TV}}(S_1,A_U)\le qb_U,\qquad d_{\mathrm{TV}}(X_{1,1},Z_U)\le qb_U,\qquad q=\gamma_1^2<225. \tag*{(27)} d TV ( S 1 , A U ) ≤ q b U , d TV ( X 1 , 1 , Z U ) ≤ q b U , q = γ 1 2 < 225. (27) Both bounds are less than 4.5 ⋅ 10 − 9 4.5\cdot10^{-9} 4.5 ⋅ 1 0 − 9 U = T ∗ U=T_\ast U = T ∗ P F ∞ \mathrm{PF}_{\infty} PF ∞
Proof. The counting bound gives
N ( U / 3 ) − N ( U / 4 ) ≥ U 24 π log U 8 π − 1 4 log ( U + 2 ) − 8 > 0 ( U ≥ 1000 ) . N(U/3)-N(U/4)\ge\frac{U}{24\pi}\log\frac{U}{8\pi}-\frac14\log(U+2)-8>0\qquad(U\ge1000). N ( U /3 ) − N ( U /4 ) ≥ 24 π U log 8 π U − 4 1 log ( U + 2 ) − 8 > 0 ( U ≥ 1000 ) . Indeed, the first term exceeds U / 32 U/32 U /32 γ ∗ \gamma_\ast γ ∗ Lemma 2.2 . Differentiating (C.3) at zero gives
E R U = γ ∗ − 2 + ∑ γ > U Re 1 λ ρ ≤ 16 U 2 + ∑ γ > U γ − 2 ≤ b U , \mathbb{E}R_U=\gamma_\ast^{-2}+\sum_{\gamma>U}\operatorname{Re}\frac{1}{\lambda_\rho}\le\frac{16}{U^2}+\sum_{\gamma>U}\gamma^{-2}\le b_U, E R U = γ ∗ − 2 + γ > U ∑ Re λ ρ 1 ≤ U 2 16 + γ > U ∑ γ − 2 ≤ b U , since Re ( λ ρ − 1 ) = ( γ 2 − δ ρ 2 ) / ( γ 2 + δ ρ 2 ) 2 ≤ γ − 2 \operatorname{Re}(\lambda_\rho^{-1})=(\gamma^2-\delta_\rho^2)/(\gamma^2+\delta_\rho^2)^2\le\gamma^{-2} Re ( λ ρ − 1 ) = ( γ 2 − δ ρ 2 ) / ( γ 2 + δ ρ 2 ) 2 ≤ γ − 2
The finite sums include the first-zero factors. Translation by r ≥ 0 r\ge0 r ≥ 0 Exp ( q ) \operatorname{Exp}(q) Exp ( q ) 1 − e − q r ≤ q r 1-e^{-qr}\le qr 1 − e − q r ≤ q r R U R_U R U l ( x ) = γ 1 e − γ 1 ∣ x ∣ / 2 l(x)=\gamma_1e^{-\gamma_1|x|}/2 l ( x ) = γ 1 e − γ 1 ∣ x ∣ /2 D 2 l = q l − q δ 0 D^2l=ql-q\delta_0 D 2 l = q l − q δ 0 2 q 2q 2 q
∥ l ( ⋅ − y ) + l ( ⋅ + y ) − 2 l ∥ 1 ≤ 2 q y 2 . \lVert l(\mathord{\cdot}-y)+l(\mathord{\cdot}+y)-2l\rVert_1\le2qy^2. ∥ l ( ⋅ − y ) + l ( ⋅ + y ) − 2 l ∥ 1 ≤ 2 q y 2 . This follows by integrating translates of D 2 l D^2l D 2 l ( ∣ y ∣ − ∣ v ∣ ) + (|y|-|v|)_+ ( ∣ y ∣ − ∣ v ∣ ) + Y U Y_U Y U E Y U 2 = 2 E R U \mathbb{E}Y_U^2=2\mathbb{E}R_U E Y U 2 = 2 E R U L 1 L^1 L 1 d T V ( L γ 1 + Y U , L γ 1 ) ≤ q E R U d_{\mathrm{TV}}(L_{\gamma_1}+Y_U,L_{\gamma_1})\le q\mathbb{E}R_U d TV ( L γ 1 + Y U , L γ 1 ) ≤ q E R U q < 225 q<225 q < 225 U = T ∗ U=T_\ast U = T ∗
These estimates concern the normalized densities c 0 Θ c_0\Theta c 0 Θ c 0 Λ c_0\Lambda c 0 Λ
Exponential moments of the residual
For order ten we use the much smaller residual obtained by retaining a zero near the verified height. The next bound controls the whole residual law and is the input to the finite determinant certificate.
Lemma C.5. Take U = T ∗ U=T_\ast U = T ∗ γ ∗ ∈ ( U / 4 , U / 3 ] \gamma_\ast\in(U/4,U/3] γ ∗ ∈ ( U /4 , U /3 ] b = b U b=b_U b = b U Ω = 10 5 \Omega=10^5 Ω = 1 0 5 Y U = 2 R U G 1 Y_U=\sqrt{2R_U}G_1 Y U = 2 R U G 1
E Y U 2 ≤ 2 b , E cosh ( Ω Y U ) − 1 ≤ e 2 Ω 2 b − 1 < 1 2 . \mathbb{E}Y_U^2\le2b,\qquad\mathbb{E}\cosh(\Omega Y_U)-1\le e^{2\Omega^2b}-1<\frac12. E Y U 2 ≤ 2 b , E cosh ( Ω Y U ) − 1 ≤ e 2 Ω 2 b − 1 < 2 1 . Proof. The variance bound follows from E R U ≤ b \mathbb{E}R_U\le b E R U ≤ b 0 ≤ v ≤ Ω 2 0\le v\le\Omega^2 0 ≤ v ≤ Ω 2
∫ 0 ∞ e v t K U ( t ) d t ≤ 1 γ ∗ 2 − v + ∑ γ > U 1 γ 2 − 1 / 4 − v ≤ 2 b . \int_0^\infty e^{vt}K_U(t)\,dt\le\frac{1}{\gamma_\ast^2-v}+\sum_{\gamma>U}\frac{1}{\gamma^2-1/4-v}\le2b. ∫ 0 ∞ e v t K U ( t ) d t ≤ γ ∗ 2 − v 1 + γ > U ∑ γ 2 − 1/4 − v 1 ≤ 2 b . Indeed, Ω 2 ≤ U 2 / 32 < γ ∗ 2 / 2 \Omega^2 \le U^2/32 < \gamma_*^2/2 Ω 2 ≤ U 2 /32 < γ ∗ 2 /2 Ω 2 + 1 / 4 ≤ U 2 / 2 \Omega^2 + 1/4 \le U^2/2 Ω 2 + 1/4 ≤ U 2 /2
log E e Ω 2 R U = ∫ 0 Ω 2 ∫ 0 ∞ e v t K U ( t ) d t d v ≤ 2 Ω 2 b . \log\mathbb{E}e^{\Omega^2 R_U} = \int_0^{\Omega^2}\int_0^\infty e^{vt}K_U(t)\,dt\,dv \le2\Omega^2 b. log E e Ω 2 R U = ∫ 0 Ω 2 ∫ 0 ∞ e v t K U ( t ) d t d v ≤ 2 Ω 2 b . Gaussian conditioning yields the cosh bound. Appendix § 10 .7 certifies the final scalar inequality. □
The order PF10 for Lambda
Log-concavity controls only two-by-two minors. For higher orders we use signed differential determinants,
D k ( p ; x ) = ( − 1 ) k ( k − 1 ) / 2 det [ p ( i + j ) ( x ) ] i , j = 0 k − 1 . \mathcal{D}_k(p;x)=(-1)^{k(k-1)/2}\det\left[p^{(i+j)}(x)\right]_{i,j=0}^{k-1}. D k ( p ; x ) = ( − 1 ) k ( k − 1 ) /2 det [ p ( i + j ) ( x ) ] i , j = 0 k − 1 . Their positivity gives positivity of the PFk _k k
Lemma C.6 (Differential criterion for PF minors). If p ∈ C 2 m − 2 ( R ) p \in C^{2m-2}(\mathbb{R}) p ∈ C 2 m − 2 ( R ) D k ( p ; x ) > 0 \mathcal{D}_k(p;x)>0 D k ( p ; x ) > 0 x ∈ R x\in\mathbb{R} x ∈ R 1 ≤ k ≤ m 1\le k\le m 1 ≤ k ≤ m p ( x − y ) p(x-y) p ( x − y ) m m m
Proof. Fix y y y x ↦ ( − 1 ) j p ( j ) ( x − y ) x\mapsto(-1)^j p^{(j)}(x-y) x ↦ ( − 1 ) j p ( j ) ( x − y ) 0 ≤ j < m 0\le j<m 0 ≤ j < m D k ( p ; x − y ) \mathcal{D}_k(p;x-y) D k ( p ; x − y ) Lemma C.1 makes their ordered evaluation determinants positive. For fixed increasing x i x_i x i y y y y ↦ p ( x i − y ) y\mapsto p(x_i-y) y ↦ p ( x i − y )
Lemma C.7 (Eleven Laplace factors with a small symmetric residual). Let a i ∈ I i a_i\in I_i a i ∈ I i 1 ≤ i ≤ 11 1\le i\le11 1 ≤ i ≤ 11 I i I_i I i § 10 .6. Their centers are approximately
14.13 , 21.02 , 32.94 , 48.01 , 72.07 , 107.17 , 14.13,\ 21.02,\ 32.94,\ 48.01,\ 72.07,\ 107.17, 14.13 , 21.02 , 32.94 , 48.01 , 72.07 , 107.17 , 161.19 , 241.05 , 361.29 , 544.32 , 814.87. 161.19,\ 241.05,\ 361.29,\ 544.32,\ 814.87. 161.19 , 241.05 , 361.29 , 544.32 , 814.87. L a 1 , … , L a 11 , Y L_{a_1},\ldots,L_{a_{11}},Y L a 1 , … , L a 11 , Y L a i ∼ Lap ( a i ) L_{a_i}\sim\operatorname{Lap}(a_i) L a i ∼ Lap ( a i ) Y Y Y
E Y 2 ≤ 2 b , E cosh ( Ω Y ) − 1 ≤ κ : = 1 2 , Ω = 10 5 , b = b T ∗ . \mathbb{E}Y^2\le2b,\qquad\mathbb{E}\cosh(\Omega Y)-1\le\kappa:=\frac{1}{2},\qquad\Omega=10^5,\qquad b=b_{T_*}. E Y 2 ≤ 2 b , E cosh ( Ω Y ) − 1 ≤ κ := 2 1 , Ω = 1 0 5 , b = b T ∗ . The density p p p Y + ∑ i = 1 11 L a i Y+\sum_{i=1}^{11}L_{a_i} Y + ∑ i = 1 11 L a i D k ( p ; x ) > 0 \mathcal{D}_k(p;x)>0 D k ( p ; x ) > 0 x ∈ R x\in\mathbb{R} x ∈ R 1 ≤ k ≤ 10 1\le k\le10 1 ≤ k ≤ 10
Proof. Write f f f
f ( x ) = ∑ i = 1 11 c i e − a i ∣ x ∣ , c i = ∏ j = 1 11 a j 2 2 a i ∏ j ≠ i ( a j 2 − a i 2 ) . f(x)=\sum_{i=1}^{11}c_i e^{-a_i|x|},\qquad c_i=\frac{\prod_{j=1}^{11}a_j^2}{2a_i\prod_{j\ne i}(a_j^2-a_i^2)}. f ( x ) = i = 1 ∑ 11 c i e − a i ∣ x ∣ , c i = 2 a i ∏ j = i ( a j 2 − a i 2 ) ∏ j = 1 11 a j 2 . The coefficients alternate in sign, starting positive. The function is C 20 C^{20} C 20 S = 100 S=100 S = 100 r i = a i / S r_i=a_i/S r i = a i / S C j = ∑ i ∣ c i ∣ r i j C_j=\sum_i |c_i|r_i^j C j = ∑ i ∣ c i ∣ r i j 0 ≤ j ≤ 18 0\le j\le18 0 ≤ j ≤ 18
p ( j ) ( x ) = E f ( j ) ( x − Y ) , ∣ p ( j ) ( x ) − f ( j ) ( x ) S j ∣ ≤ b S 2 C j + 2 . (28) p^{(j)}(x)=\mathbb{E}f^{(j)}(x-Y),\qquad\left|\frac{p^{(j)}(x)-f^{(j)}(x)}{S^j}\right|\le bS^2C_{j+2}. \tag*{(28)} p ( j ) ( x ) = E f ( j ) ( x − Y ) , S j p ( j ) ( x ) − f ( j ) ( x ) ≤ b S 2 C j + 2 . (28) In particular there is no loss of derivative order at zero.
For x > 0 x>0 x > 0 M i = E e a i Y M_i=\mathbb{E}e^{a_iY} M i = E e a i Y g ( x ) = ∑ i c i M i e − a i x g(x)=\sum_i c_iM_i e^{-a_i x} g ( x ) = ∑ i c i M i e − a i x a i < Ω a_i<\Omega a i < Ω
1 ≤ M i ≤ 1 + κ ( a i / Ω ) 2 ≤ M ‾ , M ‾ = 1 + κ ( a 11 / Ω ) 2 . 1\le M_i\le1+\kappa(a_i/\Omega)^2\le\overline{M},\qquad\overline{M}=1+\kappa(a_{11}/\Omega)^2. 1 ≤ M i ≤ 1 + κ ( a i /Ω ) 2 ≤ M , M = 1 + κ ( a 11 /Ω ) 2 . Only Y > x Y>x Y > x p p p g g g cosh ( Ω Y ) − 1 ≥ 1 2 e Ω Y ( 1 − e − Ω x ) 2 \cosh(\Omega Y)-1\ge\frac{1}{2}e^{\Omega Y}(1-e^{-\Omega x})^2 cosh ( Ω Y ) − 1 ≥ 2 1 e Ω Y ( 1 − e − Ω x ) 2
∣ p ( j ) ( x ) − g ( j ) ( x ) S j ∣ ≤ C j d ( x ) , d ( x ) = 4 κ e − Ω x ( 1 − e − Ω x ) 2 , 0 ≤ j ≤ 18. (29) \left|\frac{p^{(j)}(x)-g^{(j)}(x)}{S^j}\right|\le C_jd(x),\qquad d(x)=\frac{4\kappa e^{-\Omega x}}{(1-e^{-\Omega x})^2},\qquad0\le j\le18. \tag*{(29)} S j p ( j ) ( x ) − g ( j ) ( x ) ≤ C j d ( x ) , d ( x ) = ( 1 − e − Ω x ) 2 4 κ e − Ω x , 0 ≤ j ≤ 18. (29) The matching derivatives of f f f p = f ∗ L ( Y ) p=f\ast L(Y) p = f ∗ L ( Y ) Y = x Y=x Y = x
For 0 ≤ u , v < k 0\le u,v<k 0 ≤ u , v < k G u v = g ( u + v ) ( x ) / S u + v G_{uv}=g^{(u+v)}(x)/S^{u+v} G uv = g ( u + v ) ( x ) / S u + v E u v = ( p ( u + v ) − g ( u + v ) ) ( x ) / S u + v E_{uv}=(p^{(u+v)}-g^{(u+v)})(x)/S^{u+v} E uv = ( p ( u + v ) − g ( u + v ) ) ( x ) / S u + v
( − 1 ) k ( k − 1 ) / 2 det G = ∑ ∣ I ∣ = k w I ( ∏ i ∈ I M i ) e − S I x , S I = ∑ i ∈ I a i , w I = ( − 1 ) k ( k − 1 ) / 2 ∏ i ∈ I c i ∏ i < j i , j ∈ I ( r j − r i ) 2 . (-1)^{k(k-1)/2}\det G=\sum_{|I|=k}w_I\left(\prod_{i\in I}M_i\right)e^{-S_Ix},\qquad S_I=\sum_{i\in I}a_i,\qquad w_I=(-1)^{k(k-1)/2}\prod_{i\in I}c_i\prod_{\substack{i<j\\i,j\in I}}(r_j-r_i)^2. ( − 1 ) k ( k − 1 ) /2 det G = ∣ I ∣ = k ∑ w I ( i ∈ I ∏ M i ) e − S I x , S I = i ∈ I ∑ a i , w I = ( − 1 ) k ( k − 1 ) /2 i ∈ I ∏ c i i < j i , j ∈ I ∏ ( r j − r i ) 2 . For I 0 = { 1 , … , k } I_0=\{1,\ldots,k\} I 0 = { 1 , … , k } w 0 = w I 0 > 0 w_0=w_{I_0}>0 w 0 = w I 0 > 0 S 0 = S I 0 S_0=S_{I_0} S 0 = S I 0 C j ≤ C 0 r 11 j C_j\le C_0r_{11}^j C j ≤ C 0 r 11 j ∣ G u v ∣ ≤ C 0 r 11 u + v M ‾ e − a 1 x |G_{uv}|\le C_0r_{11}^{u+v}\overline{M}e^{-a_1x} ∣ G uv ∣ ≤ C 0 r 11 u + v M e − a 1 x
∣ det ( G + E ) − det G ∣ w 0 e − S 0 x ≤ E k ( x ) : = k ! C 0 k r 11 k ( k − 1 ) w 0 ∑ j = 1 k ( k j ) M ‾ k − j ( 4 κ ) j e − [ j Ω + ( k − j ) a 1 − S 0 ] x ( 1 − e − Ω x ) 2 j . (30) \frac{|\det(G+E)-\det G|}{w_0e^{-S_0x}}\le\mathcal{E}_k(x):=\frac{k!C_0^kr_{11}^{k(k-1)}}{w_0}\sum_{j=1}^{k}\binom{k}{j}\overline{M}^{k-j}(4\kappa)^j\frac{e^{-[j\Omega+(k-j)a_1-S_0]x}}{(1-e^{-\Omega x})^{2j}}. \tag*{(30)} w 0 e − S 0 x ∣ det ( G + E ) − det G ∣ ≤ E k ( x ) := w 0 k ! C 0 k r 11 k ( k − 1 ) j = 1 ∑ k ( j k ) M k − j ( 4 κ ) j ( 1 − e − Ω x ) 2 j e − [ j Ω + ( k − j ) a 1 − S 0 ] x . (30) All exponent differences on the right are positive for all a i ∈ I i a_i\in I_i a i ∈ I i ( 0 , ∞ ) (0,\infty) ( 0 , ∞ )
Appendix § 10 .7 certifies the remaining finite inequalities. On [ 0 , 1 / 250 ] [0,1/250] [ 0 , 1/250 ] [ 1 / 250 , 1 / 10 ] [1/250,1/10] [ 1/250 , 1/10 ] E k ( x ) \mathcal{E}_k(x) E k ( x ) 2 ≤ k ≤ 10 2\le k\le10 2 ≤ k ≤ 10 x ≥ 1 / 10 x\ge1/10 x ≥ 1/10 0.18 0.18 0.18 1 / 10 1/10 1/10 E k \mathcal{E}_k E k 1 / 10 1/10 1/10 p p p x < 0 x<0 x < 0
Proof of the order-ten assertion in Theorem 1.2 . The endpoint sign certificates in Appendix § 10 .6 supply a real zero in each I i I_i I i U = T ∗ U=T_* U = T ∗ γ ∗ ∈ ( U / 4 , U / 3 ] \gamma_*\in(U/4,U/3] γ ∗ ∈ ( U /4 , U /3 ] Lemmas C.7 and C.6 give strictly positive ordered minors through order ten for these eleven factors convolved with Y U Y_U Y U
Convolution with each remaining Laplace density preserves strictness. The classical composition formula for totally positive kernels (continuous Cauchy–Binet; Karlin, [12 ] ) integrates the product of a strictly positive minor and a nonnegative Laplace minor over u 1 < ⋯ < u k u_1<\cdots<u_k u 1 < ⋯ < u k u i = y i u_i=y_i u i = y i [ e − a ∣ y i − y j ∣ ] [e^{-a|y_i-y_j|}] [ e − a ∣ y i − y j ∣ ] y i y_i y i c 0 Λ c_0\Lambda c 0 Λ Λ \Lambda Λ
Fixed-order positivity near the one-sided boundary
We use the exponential-tilting (saddlepoint) method of Daniels [10 ] to obtain relative determinant estimates near the left endpoint of Θ \Theta Θ
Proposition C.8 (Local PF positivity of Θ \Theta Θ For each fixed integer m ≥ 1 m\ge1 m ≥ 1 ϵ m > 0 \epsilon_m>0 ϵ m > 0
det [ Θ ( x i − y j ) ] i , j = 1 k > 0 ( 1 ≤ k ≤ m ) \det[\Theta(x_i-y_j)]_{i,j=1}^{k}>0\qquad(1\le k\le m) det [ Θ ( x i − y j ) ] i , j = 1 k > 0 ( 1 ≤ k ≤ m ) whenever x 1 < ⋯ < x k x_1<\cdots<x_k x 1 < ⋯ < x k y 1 < ⋯ < y k y_1<\cdots<y_k y 1 < ⋯ < y k 0 < x i − y j < ϵ m 0<x_i-y_j<\epsilon_m 0 < x i − y j < ϵ m i , j i,j i , j x s = B ′ ( s ) x_s=B'(s) x s = B ′ ( s ) v s = − B ′ ′ ( s ) v_s=-B''(s) v s = − B ′′ ( s ) s → ∞ s\to\infty s → ∞
D k ( Θ ; x s ) ∼ Θ ( x s ) k v s − k ( k − 1 ) / 2 ∏ j = 0 k − 1 j ! (31) \mathcal{D}_k(\Theta;x_s)\sim\Theta(x_s)^kv_s^{-k(k-1)/2}\prod_{j=0}^{k-1}j! \tag*{(31)} D k ( Θ ; x s ) ∼ Θ ( x s ) k v s − k ( k − 1 ) /2 j = 0 ∏ k − 1 j ! (31) Proof. We use the branch of B ( z ) = log h ( z ) B(z)=\log h(z) B ( z ) = log h ( z ) [19 Section 5.11] , together with log ζ ( w ) = O ( 2 − Re w ) \log\zeta(w)=O(2^{-\operatorname{Re} w}) log ζ ( w ) = O ( 2 − Re w ) Re w → ∞ \operatorname{Re} w\to\infty Re w → ∞
B ( z ) = z 2 ( log z 2 π − 1 ) + 7 8 log z + C + O ( ∣ z ∣ − 1 / 2 ) (32) B(z)=\frac{\sqrt{z}}{2}\left(\log\frac{\sqrt{z}}{2\pi}-1\right)+\frac{7}{8}\log z+C+O(|z|^{-1/2}) \tag*{(32)} B ( z ) = 2 z ( log 2 π z − 1 ) + 8 7 log z + C + O ( ∣ z ∣ − 1/2 ) (32) uniformly in the right half-plane as ∣ z ∣ → ∞ |z|\to\infty ∣ z ∣ → ∞ ∣ arg z ∣ ≤ π / 4 |\arg\sqrt{z}|\leq\pi/4 ∣ arg z ∣ ≤ π /4 Re z ≥ ∣ z ∣ / 2 \operatorname{Re}\sqrt{z}\geq\sqrt{|z|}/2 Re z ≥ ∣ z ∣ /2
x s ∼ log s 8 s , v s ∼ log s 16 s 3 / 2 , B ′ ′ ′ ( z ) = O ( log s s 5 / 2 ) ( ∣ z − s ∣ ≤ s / 2 ) . (33) x_s\sim\frac{\log s}{8\sqrt{s}},\qquad v_s\sim\frac{\log s}{16s^{3/2}},\qquad B'''(z)=O\left(\frac{\log s}{s^{5/2}}\right)\quad(|z-s|\leq s/2). \tag*{(33)} x s ∼ 8 s log s , v s ∼ 16 s 3/2 log s , B ′′′ ( z ) = O ( s 5/2 log s ) ( ∣ z − s ∣ ≤ s /2 ) . (33) For large s s s v s > 0 v_s>0 v s > 0 x s x_s x s
The tilted density is f s ( u ) = e B ( s ) − s u θ 1 ( u ) f_s(u)=e^{B(s)-su}\theta_1(u) f s ( u ) = e B ( s ) − s u θ 1 ( u ) x s x_s x s v s v_s v s σ s = v s \sigma_s=\sqrt{v_s} σ s = v s g s ( z ) = σ s f s ( x s + σ s z ) g_s(z)=\sigma_s f_s(x_s+\sigma_s z) g s ( z ) = σ s f s ( x s + σ s z )
ϕ s ( t ) = exp { B ( s ) − B ( s − i t / σ s ) − i t x s / σ s } . \phi_s(t)=\exp\{B(s)-B(s-it/\sigma_s)-itx_s/\sigma_s\}. ϕ s ( t ) = exp { B ( s ) − B ( s − i t / σ s ) − i t x s / σ s } . Taylor expansion using (C.10) gives ϕ s ( t ) → e − t 2 / 2 \phi_s(t)\to e^{-t^2/2} ϕ s ( t ) → e − t 2 /2 t t t
To pass to density derivatives, we need an integrable bound for all frequencies. Write 1 + i u = a + i b \sqrt{1+iu}=a+ib 1 + i u = a + ib a ≥ 1 a\geq1 a ≥ 1 w = s w=\sqrt{s} w = s L = log ( w / ( 2 π ) ) − 1 L=\log(w/(2\pi))-1 L = log ( w / ( 2 π )) − 1
Re B ( s + i s u ) − B ( s ) ≥ w L 2 ( a − 1 ) − C 1 w . \operatorname{Re}B(s+isu)-B(s)\geq\frac{wL}{2}(a-1)-\frac{C_1}{w}. Re B ( s + i s u ) − B ( s ) ≥ 2 w L ( a − 1 ) − w C 1 . Here Re [ ( a + i b ) log ( a + i b ) ] ≥ 0 \operatorname{Re}[(a+ib)\log(a+ib)]\geq0 Re [( a + ib ) log ( a + ib )] ≥ 0 η = ∣ b ∣ / a < 1 \eta=|b|/a<1 η = ∣ b ∣/ a < 1 a a a arctanh ( η 2 ) − η arctan η \operatorname{arctanh}(\eta^2)-\eta\arctan\eta arctanh ( η 2 ) − η arctan η η / ( 1 − η 2 ) − arctan η ≥ 0 \eta/(1-\eta^2)-\arctan\eta\geq0 η / ( 1 − η 2 ) − arctan η ≥ 0 7 8 log ∣ 1 + i u ∣ \frac{7}{8}\log|1+iu| 8 7 log ∣1 + i u ∣
a − 1 = u 2 2 ( a + 1 ) ( 1 + u 2 + 1 ) ≥ c 1 min { u 2 , ∣ u ∣ } a-1=\frac{u^2}{2(a+1)(\sqrt{1+u^2}+1)}\geq c_1\min\{u^2,\sqrt{|u|}\} a − 1 = 2 ( a + 1 ) ( 1 + u 2 + 1 ) u 2 ≥ c 1 min { u 2 , ∣ u ∣ } for an absolute c 1 > 0 c_1>0 c 1 > 0 A s = s σ s A_s=s\sigma_s A s = s σ s A s 2 ∼ w L / 8 A_s^2\sim wL/8 A s 2 ∼ w L /8 A s → ∞ A_s\to\infty A s → ∞ u = t / A s u=t/A_s u = t / A s ∣ ϕ s ( t ) ∣ ≤ C 2 exp { − c min ( t 2 , A s 3 / 2 ∣ t ∣ ) } |\phi_s(t)|\leq C_2\exp\{-c\min(t^2,A_s^{3/2}\sqrt{|t|})\} ∣ ϕ s ( t ) ∣ ≤ C 2 exp { − c min ( t 2 , A s 3/2 ∣ t ∣ )} c , C 2 > 0 c,C_2>0 c , C 2 > 0 s s s A s ≥ 1 A_s\geq1 A s ≥ 1
∣ ϕ s ( t ) ∣ ≤ C 2 exp { − c min ( t 2 , ∣ t ∣ ) } . |\phi_s(t)|\leq C_2\exp\{-c\min(t^2,\sqrt{|t|})\}. ∣ ϕ s ( t ) ∣ ≤ C 2 exp { − c min ( t 2 , ∣ t ∣ )} . The right side is integrable after multiplication by any fixed power of ∣ t ∣ |t| ∣ t ∣ g s ( r ) ( 0 ) → φ ( r ) ( 0 ) g_s^{(r)}(0)\to\varphi^{(r)}(0) g s ( r ) ( 0 ) → φ ( r ) ( 0 ) r r r φ ( z ) = ( 2 π ) − 1 / 2 e − z 2 / 2 \varphi(z)=(2\pi)^{-1/2}e^{-z^2/2} φ ( z ) = ( 2 π ) − 1/2 e − z 2 /2
D k ( φ ; 0 ) = φ ( 0 ) k ∏ j = 0 k − 1 j ! > 0. \mathcal{D}_k(\varphi;0)=\varphi(0)^k\prod_{j=0}^{k-1}j!>0. D k ( φ ; 0 ) = φ ( 0 ) k j = 0 ∏ k − 1 j ! > 0. For example, this follows by replacing monomials with monic Hermite polynomials in the Gaussian Gram matrix; their squared norms are j ! j! j ! M f = ( f ( i + j ) ( x ) ) i , j = 0 k − 1 M_f=(f^{(i+j)}(x))_{i,j=0}^{k-1} M f = ( f ( i + j ) ( x ) ) i , j = 0 k − 1 M e c x f = e c x T c M f T c T M_{e^{cx}f}=e^{cx}T_cM_fT_c^{\mathsf T} M e c x f = e c x T c M f T c T ( T c ) i j = ( i j ) c i − j (T_c)_{ij}=\binom{i}{j}c^{i-j} ( T c ) ij = ( j i ) c i − j j ≤ i j\leq i j ≤ i det T c = 1 \det T_c=1 det T c = 1 e k c x e^{kcx} e k c x
D k ( Θ ; x s ) Θ ( x s ) k = σ s − k ( k − 1 ) D k ( g s ; 0 ) g s ( 0 ) k , \frac{\mathcal{D}_k(\Theta;x_s)}{\Theta(x_s)^k}=\sigma_s^{-k(k-1)}\frac{\mathcal{D}_k(g_s;0)}{g_s(0)^k}, Θ ( x s ) k D k ( Θ ; x s ) = σ s − k ( k − 1 ) g s ( 0 ) k D k ( g s ; 0 ) , which proves (C.8). All initial signed determinants through size m m m ( 0 , ε m ) (0,\varepsilon_m) ( 0 , ε m ) ε m > 0 \varepsilon_m>0 ε m > 0 P F k \mathrm{PF}_k PF k
The eleven real-zero enclosures
Define I i = [ a ˉ i − 10 − 40 , a ˉ i + 10 − 40 ] I_i=[\bar a_i-10^{-40},\bar a_i+10^{-40}] I i = [ a ˉ i − 1 0 − 40 , a ˉ i + 1 0 − 40 ]
Interval evaluation gives opposite signs for Ξ \Xi Ξ I i I_i I i
The evaluation uses Euler–Maclaurin with N = 1000 N=1000 N = 1000 p = 64 p=64 p = 64 s = 1 / 2 + i t s=1/2+it s = 1/2 + i t
ζ ( s ) = ∑ n = 1 N − 1 n − s + N 1 − s s − 1 + 1 2 N − s + ∑ k = 1 p B 2 k ( 2 k ) ! ( s ) 2 k − 1 N − s − 2 k + 1 + R p ( s ) , \zeta(s)=\sum_{n=1}^{N-1}n^{-s}+\frac{N^{1-s}}{s-1}+\frac{1}{2}N^{-s}+\sum_{k=1}^{p}\frac{B_{2k}}{(2k)!}(s)_{2k-1}N^{-s-2k+1}+R_p(s), ζ ( s ) = n = 1 ∑ N − 1 n − s + s − 1 N 1 − s + 2 1 N − s + k = 1 ∑ p ( 2 k )! B 2 k ( s ) 2 k − 1 N − s − 2 k + 1 + R p ( s ) , ∣ R p ( s ) ∣ ≤ 4 ∣ ( s ) 2 p ∣ ( 2 π ) 2 p ( Re s + 2 p − 1 ) N Re s + 2 p − 1 . \lvert R_p(s)\rvert\le\frac{4\lvert(s)_{2p}\rvert}{(2\pi)^{2p}(\operatorname{Re}s+2p-1)N^{\operatorname{Re}s+2p-1}}. ∣ R p ( s )∣ ≤ ( 2 π ) 2 p ( Re s + 2 p − 1 ) N Re s + 2 p − 1 4 ∣( s ) 2 p ∣ . Here ( s ) j = s ( s + 1 ) ⋯ ( s + j − 1 ) (s)_j=s(s+1)\cdots(s+j-1) ( s ) j = s ( s + 1 ) ⋯ ( s + j − 1 ) B 2 k B_{2k} B 2 k ∥ B 2 p ( { ⋅ } ) ∥ ∞ / ( 2 p ) ! ≤ 4 / ( 2 π ) 2 p \lVert B_{2p}(\{\cdot\})\rVert_\infty/(2p)!\le4/(2\pi)^{2p} ∥ B 2 p ({ ⋅ }) ∥ ∞ / ( 2 p )! ≤ 4/ ( 2 π ) 2 p s ( s − 1 ) π − s / 2 Γ ( s / 2 ) / 2 s(s-1)\pi^{-s/2}\Gamma(s/2)/2 s ( s − 1 ) π − s /2 Γ ( s /2 ) /2 Ξ ( t ) \Xi(t) Ξ ( t )
The order-ten certificate
With U = T ∗ U=T_* U = T ∗ b = b U b=b_U b = b U Ω = 10 5 \Omega=10^5 Ω = 1 0 5
Ω 2 ≤ U 2 / 32 , Ω 2 + 1 / 4 ≤ U 2 / 2 , e 2 Ω 2 b − 1 < 1 2 . \Omega^2\le U^2/32,\qquad\Omega^2+1/4\le U^2/2,\qquad e^{2\Omega^2b}-1<\frac{1}{2}. Ω 2 ≤ U 2 /32 , Ω 2 + 1/4 ≤ U 2 /2 , e 2 Ω 2 b − 1 < 2 1 . Together with the analytic mean bound E R U ≤ b \mathop{\mathrm{E}}R_U\le b E R U ≤ b Lemma C.7 . In the notation of that lemma, the normalized reference determinant is
F k ( x ) = ∑ ∣ I ∣ = k q I e − d I x , q I = w I w 0 ∏ i ∈ I M i , d I = S I − S 0 . (34) F_k(x)=\sum_{\lvert I\rvert=k}q_I e^{-d_Ix},\qquad q_I=\frac{w_I}{w_0}\prod_{i\in I}M_i,\qquad d_I=S_I-S_0. \tag*{(34)} F k ( x ) = ∣ I ∣ = k ∑ q I e − d I x , q I = w 0 w I i ∈ I ∏ M i , d I = S I − S 0 . (34) The certificate checks w 0 > 0 w_0>0 w 0 > 0 d I > 0 d_I>0 d I > 0 I ≠ I 0 I\ne I_0 I = I 0 1 ≤ M i ≤ 1 + 1 2 ( a i / Ω ) 2 1\le M_i\le1+\frac{1}{2}(a_i/\Omega)^2 1 ≤ M i ≤ 1 + 2 1 ( a i /Ω ) 2 j Ω + ( k − j ) a 1 − S 0 j\Omega+(k-j)a_1-S_0 j Ω + ( k − j ) a 1 − S 0 E k E_k E k ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Ω \Omega Ω
The compact interval. Centered Taylor enclosures reduce cancellation in the finite sums. For a box [ l , u ] [l,u] [ l , u ] z z z h h h V j ( x ) = ∑ i c i ( − r i ) j e − a i x V_j(x)=\sum_i c_i(-r_i)^j e^{-a_i x} V j ( x ) = ∑ i c i ( − r i ) j e − a i x r i = a i / S r_i=a_i/S r i = a i / S
V j ( [ l , u ] ) ⊆ V j ( z ) + S V j + 1 ( z ) [ − h , h ] + [ − A j , A j ] , A j = S 2 h 2 2 ∑ i ∣ c i ∣ r i j + 2 e − a i l . (35) V_j([l,u])\subseteq V_j(z)+SV_{j+1}(z)[-h,h]+[-A_j,A_j],\qquad A_j=\frac{S^2h^2}{2}\sum_i\lvert c_i\rvert r_i^{j+2}e^{-a_i l}. \tag*{(35)} V j ([ l , u ]) ⊆ V j ( z ) + S V j + 1 ( z ) [ − h , h ] + [ − A j , A j ] , A j = 2 S 2 h 2 i ∑ ∣ c i ∣ r i j + 2 e − a i l . (35) Add the symmetric interval of radius b S 2 C j + 2 bS^{2}C_{j+2} b S 2 C j + 2 0 ≤ j ≤ 18 0 \le j \le18 0 ≤ j ≤ 18 [ 0 , 1 / 250 ] [0,1/250] [ 0 , 1/250 ] ( − 1 ) j (-1)^j ( − 1 ) j j = 0 , … , 9 j=0,\ldots,9 j = 0 , … , 9 [ 1 / 250 , 1 / 10 ] [1/250,1/10] [ 1/250 , 1/10 ]
F k ( [ l , u ] ) ⊆ F k ( z ) + F k ′ ( z ) [ − h , h ] + [ − A , A ] , A = h 2 2 ∑ ∣ I ∣ = k ∣ q I ∣ d I 2 e − d I l , F_k([l,u]) \subseteq F_k(z) + F'_k(z)[-h,h] + [-A,A], \qquad A = \frac{h^2}{2}\sum_{\lvert I\rvert=k}\lvert q_I\rvert d_I^2 e^{-d_I l}, F k ([ l , u ]) ⊆ F k ( z ) + F k ′ ( z ) [ − h , h ] + [ − A , A ] , A = 2 h 2 ∣ I ∣ = k ∑ ∣ q I ∣ d I 2 e − d I l , and accept a box only if its lower bound exceeds the upper bound of E k ( l ) E_k(l) E k ( l ) 2 ≤ k ≤ 10 2 \le k \le10 2 ≤ k ≤ 10
Table 1 (PDF p. 25).
The closed boxes cover both endpoints without gaps, and each enclosure holds throughout its box, so the result is uniform rather than a grid test. Symmetry supplies the negative half-line.
The unbounded tails. For x ≥ 1 / 10 x \ge1/10 x ≥ 1/10 F k F_k F k 1 / 10 1/10 1/10
F k ( x ) ≥ m k : = 1 − ∑ w I < 0 ∣ w I ∣ w 0 ( ∏ i ∈ I M ‾ i ) e − ( S I − S 0 ) / 10 , M ‾ i = 1 + 1 2 ( a i / Ω ) 2 . F_k(x) \ge m_k := 1 - \sum_{w_I<0}\frac{\lvert w_I\rvert}{w_0}\left(\prod_{i\in I}\overline{M}_i\right)e^{-(S_I-S_0)/10}, \qquad\overline{M}_i = 1 + \frac{1}{2}(a_i/\Omega)^2. F k ( x ) ≥ m k := 1 − w I < 0 ∑ w 0 ∣ w I ∣ ( i ∈ I ∏ M i ) e − ( S I − S 0 ) /10 , M i = 1 + 2 1 ( a i /Ω ) 2 . For each 2 ≤ k ≤ 10 2 \le k \le10 2 ≤ k ≤ 10 m k > 0.18 m_k > 0.18 m k > 0.18 E k ( 1 / 10 ) / m k < 10 − 3 E_k(1/10)/m_k < 10^{-3} E k ( 1/10 ) / m k < 1 0 − 3 10 − 1000 10^{-1000} 1 0 − 1000 1 / 10 1/10 1/10
